topological vector bundle

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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
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[[!include topology - contents]]
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#### Bundles#
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[[!include bundles - contents]]
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#### Linear algebra
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[[!include homotopy - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A _topological vector bundle_ is a [[vector bundle]] in the context of _[[topology]]_: a continuously varying collection of [[vector space]] over a given [[topological space]].

For more survey and motivation see at _[[vector bundle]]_. Here we discuss the details of the general concept in [[topology]].
See also _[[differentiable vector bundle]]_ and _[[algebraic vector bundle]]_.

## Definition

We first give the more abstract definiton
in terms of slice categories (def. \ref{TopologicalVectorBundleInTermsOfSliceCategories} below) and then unwind this to the traditional definition (def \ref{TopologicalVectorBundle} below).

In the following

* $k$ is either the [[topological field]]

  * $k = \mathbb{R}$ of [[real numbers]]

  * or $k = \mathbb{C}$ of [[complex numbers]]

  equipped with the [[Euclidean space|Euclidean]] [[metric topology]].

* _[[vector space]]_ means _[[finite dimensional vector space]]_.

### In terms of slice categories
 {#InTermsOfSliceCategories}

+-- {: .num_defn #TopologicalVectorBundleInTermsOfSliceCategories}
###### Definition
**(topological vector bundles in terms of slice categories)**

Write [[Top]] for the [[category]] of [[topological spaces]], and for $X \in Top$ a space, write $Top_{/X}$ for its [[slice category]] over $X$. The [[Cartesian product]] in $Top_{/X}$ is the [[fiber product]] over $X$ in $Top$, which we denote by $(-) \times_X (-)$.
Observe $[X \times k \to X] \in Top_{/X}$ is canonically a [[field]] [[internalization|internal]] to $Top_{/X}$

A _topological vector bundle_ over $X \in Top$ is

1. an [[object ]] $[E \overset{\pi}{\to} X]$ of $Top_{/X}$

1. with the [[structure]] of an $X \times k$-[[vector space]]-object [[internalization|internal]] to $Top_{/X}$, hence

   1. a [[morphism]] $ (-)+(-) \;\colon\; E \times_X E \to E$

   1. a morphism $(-)\cdot(-) \;\colon\;  k \times E \to E$

   which satisfy the vector space axioms

such that

* ([[local trivialization|local triviality]]) there exists

  1. an [[open cover]] $\{U_i \subset X\}_{i \in I}$, regarded via the [[disjoint union space]] $U \coloneqq \underset{i \in I}{\sqcup} U_i$ of the patches as the object $ [U \to X] \in Top_{/X}$,

  1. an [[isomorphism]] of vector space objects in $Top_{/U}$

     $$
       U \times \mathbb{R}^n
         \overset{\simeq}{\longrightarrow}
       U \times_X E
       \,,
     $$

     for some $n \in \mathbb{N}$, where  $[U \times k^n \overset{pr_1}{\to} X]$ and $[U \times_X E \overset{pr_1}{\to} U]$ are regarded as a vector space objects in $Top_{U}$ in the canonical way.

It follows that $n \in \mathbb{N}$ is constant on [[connected components]]. Often this is required to be constant on all of $X$ and then called the _[[rank]]_ of the vector bundle.

A _[[homomorphism]]_ of topological vector bundles is simple a homomorphism of vector space objects in $Top_{/X}$.

Topological vector bundles over $X$ and homomorphisms between them constitutes a [[category]], usually denoted [[Vect(X)]].

=--

Notice that viewed in [[Top]], the last condition means that there is a [[diagram]] of the form

$$
  \array{
    U \times k^n
      &\overset{\simeq}{\longrightarrow}&
    U \times_X E
      &\overset{}{\longrightarrow}&
    E
    \\
    & \searrow & \downarrow &(pb)& \downarrow^{\mathrlap{\pi}}
    \\
    && U &\longrightarrow& X
  }
$$

where the square is a [[pullback square]] and the [[homeomorphism]] in the top left is fiber-wise linear.

If we say this yet more explicitly, it yields the definition as found in the traditional textbooks:

### In components

+-- {: .num_defn #TopologicalVectorBundle}
###### Definition
**(topological vector bundle in components)**

Let $X$ be a [[topological space]]. Then a _topological vector bundle_ over $X$ is

1. a [[topological space]] $E$;

1. a [[continuous function]] $E \overset{\pi}{\to} X$

1. for each $x \in X$ the stucture of a [[finite-dimensional vector space|finite-dimensional]] $k$-[[vector space]] on the [[pre-image]]

   $$E_x \coloneqq \pi^{-1}(\{x\}) \subset E$$

such that this is [[local trivialization|locally trivial]] in that there exists

1. an [[open cover]] $\{U_i \subset X\}_{i \in I}$,

1. for each $i \in I$ an $n_i \in \mathbb{N}$ and a [[homeomorphism]]

   $$
     \phi_i \;\colon\; U_i \times k^{n_i} \overset{\simeq}{\longrightarrow} \pi^{-1}(U_i) \subset E
   $$

   from the [[product topological space]] of $U_i$ with the [[real numbers]] (equipped with their [[Euclidean space]] [[metric topology]]) to the restriction of $E$ over $U_i$, such that

   1. $\phi_i$ is a map over $U_i$ in that $\pi \circ \phi_i = pr_1$, hence in that $\phi_i(\{x\} \times k^{n_i}) \subset \pi^{-1}(\{x\})$

   1. $\phi_i$ is a [[linear map]] in each fiber in that

      $$
        \underset{x \in U_i}{\forall}
        \left(
          \phi_i(x) \;\colon\;
             k^{n_i} \overset{\text{linear}}{\longrightarrow}
            E_x = \pi^{-1}(\{x\})
        \right)
        \,.
      $$

Often, but not always, it is required that the numbers $n_i$ are all equal
to some $n \in \mathbb{N}$, for
all $i \in I$, hence that the vector space fibers all have the same [[dimension]].
In this case one says that the vector bundle has _[[rank of a vector bundle|rank]]_ $n$.
(Over a [[connected topological space]] this is automatic, but the fiber dimension
may be distinct over distinct [[connected components]].)

For $[E_1 \overset{\pi_1}{\to} X]$ and $[E_2 \overset{\pi_2}{\to} X]$ two topological vector bundles over the same base space, then a _[[homomorphism]]_
between them is

* a [[continuous function]] $f \colon E_1 \longrightarrow E_2$

such that

1. $f$ respects the [[projections]]: $\pi_2 \circ f = \pi_1$;

1. for each $x \in X$ we have that $f|_x \colon (E_1)_x \to (E_2)_x$ is a [[linear map]].

=--

+-- {: .num_remark #TopologicalVectorBundlesCategory}
###### Remark
**([[Vect(X)|category of topological vector bundles]])**

For $X$ a [[topological space]], there is the [[category]] whose

* [[objects]] are the topological vector bundles over $X$,

* [[morphisms]] are the topological vector bundle homomorphisms

according to def. \ref{TopologicalVectorBundle}. This category usually denoted [[Vect(X)]].

We write $Vect(X)_{/\sim}$ for the [[set]] of [[isomorphism classes]] of this category.

=--

\begin{remark}
\label{FiberwiseOperations}
**(fiberwise operations)**
\linebreak
The [[category]] [[FinDimVect]] of [[finite dimensional vector spaces]] over a [[topological field|topological]] [[ground field]] is canonically a [[Top]]-[[enriched category]], and so are hence its [[product categories]] $FinDimVect^{n}$, for $n \in \mathbb{N}$. Any [[Top]]-[[enriched functor]]

$$
  F
  \;\colon\;
  FinDimVect^n
   \longrightarrow
  FinDimVect
$$

induces a functorial construction of new topological vector bundles $\widehat{F}(\mathcal{V}_1,, \cdots, \mathcal{V}_n)$ from any [[n-tuple]] $(\mathcal{V}_1, \mathcal{V}_2 , \cdots, \mathcal{V}_n)$ of vector bundles over the same base space $B$, by taking the new [[fiber]] over a point $b \in B$ to be (e.g. [Milnor & Stasheff 1974, p. 32](#MilnorStasheff74)):

$$
  F
  \big(
    \mathcal{V}_1,
    \cdots
    \mathcal{V}_n
  \big)
  _b
  \;\coloneqq\;
  F
  \big(
    (\mathcal{V}_1)_b,
    \cdots,
    (\mathcal{V}_n)_b
  \big)
  \,.
$$

For example:

* if $F \,\coloneqq\, (-)^\ast \,\colon\, FinDimVect \longrightarrow FinDimVect$ is the operation of forming [[dual vector spaces]], then $\widehat{F}$ constructs the fiberwise [[dual vector bundle]];

* if $F \,\coloneqq\, det \,\colon\, FinDimVect \longrightarrow FinDimVect$ is the operation of forming [[determinants]], then $\widehat{F}$ is the construction of fiberwise [[determinant line bundles]];

* if $F \,\coloneqq\, \oplus \,\colon\, FinDimVect^2 \longrightarrow FinDimVect$ is the [[direct sum]] of vector space, then $\widehat{F}$ constructs the fiberwise [[direct sum of vector bundles]] ("[[Whitney sum]]");

* if $F \,\coloneqq\, \otimes \,\colon\, FinDimVect^2 \longrightarrow FinDimVect$ is the [[tensor product of vector spaces]], then $\widehat{F}$ constructs the fiberwise [[tensor product of vector bundles]].

\end{remark}

+-- {: .num_remark #TerminologyVectorBundles}
###### Remark
**(some terminology)**

Let $k$ and $n$ be as in def. \ref{TopologicalVectorBundle}. Then:

For $k = \mathbb{R}$ one speaks of _[[real vector bundles]]_.

For $k = \mathbb{C}$ one speaks of _[[complex vector bundles]]_.

For $n = 1$ one speaks of _[[line bundles]]_,
in particular of _[[real line bundles]]_ and of _[[complex line bundles]]_.

=--

+-- {: .num_remark #CommonOpenCoverLocalTrivialization}
###### Remark
**(any two topologial vector bundles have [[local trivialization]] over a common [[open cover]])**

Let $[E_1 \to X]$ and $[E_2 \to X]$ be two topological vector bundles (def. \ref{TopologicalVectorBundle}).
Then there always exists an [[open cover]] $\{U_i \subset X\}_{i \in I}$ such that both bundles
have a [[local trivialization]] over this cover.

=--

+-- {: .proof}
###### Proof

By definition we may find two possibly different
open covers $\{U^1_{i_1} \subset X\}_{{i_1} \in I_1}$ and $\{U^2_{i_2} \subset X\}_{i_2 \in I_2}$
with local tivializations
$\{ U^1_{i_1} \underoverset{\simeq}{\phi^1_{i_1}}{\to} E_1\vert_{U^1_{i_1}} \}_{i_1 \in I_1}$
and
$\{ U^2_{i_2} \underoverset{\simeq}{\phi^2_{i_2}}{\to} E_2\vert_{U^2_{i_2}} \}_{i_2 \in I_2}$.

The _joint [[refinement]]_ of these two covers is the open cover

$$
  \left\{
    U_{i_1, i_2}
      \coloneqq
    U^1_{i_1} \cap U^2_{i_2}
      \subset
    X
  \right\}_{(i_1, i_2) \in I_1 \times I_2}
  \,.
$$

The original local trivializations restrict to local trivializations on this finer cover

$$
  \left\{
    U_{i_1, i_2}
      \underoverset{\simeq}{\phi^1_{i_1}\vert_{U^2_{i_2}}}{\longrightarrow}
    E_1\vert_{U_{i_1, i_2}}
  \right\}_{(i_1, i_2) \in I_1 \times I_2}
$$

and

$$
  \left\{
    U_{i_1, i_2}
      \underoverset{\simeq}{\phi^2_{i_2}\vert_{U^1_{i_1}}}{\longrightarrow}
    E_2\vert_{U_{i_1, i_2}}
  \right\}_{(i_1, i_2) \in I_1 \times I_2}
  \,.
$$

=--

+-- {: .num_example #TrivialTopologicalVectorBundle}
###### Example
**(trivial topological vector bundle and (local) trivialization)**

For $X$ any [[topological space]], and $n \in \mathbb{N}$,
we have that the [[product topological space]]

$$
  X \times k^n \overset{pr_1}{\to} X
$$

canonically becomes a topological vector bundle over $X$ (def. \ref{TopologicalVectorBundle}). This is called the _[[trivial vector bundle]]_ of [[rank]] $n$ over $X$.

Given any topological vector bundle $E \to X$, then a choice of [[isomorphism]] to a trivial bundle (if it exists)

$$
  E \overset{\simeq}{\longrightarrow} X \times k^n
$$

is called a _trivialization_ of $E$. A vector bundle for which a trivialization exists is called _trivializable_.

Accordingly, the [[local trivialization|local triviality]] condition in the definition of topological vector bundles (def. \ref{TopologicalVectorBundle}) says that they are locally isomorphic to the trivial vector bundle. One also says that the data consisting of an open cover $\{U_i \subset X\}_{i \in I}$ and the [[homeomorphisms]]

$$
  \left\{
     U_i \times k^n
       \overset{\simeq}{\to}
     E|_{U_i}
  \right\}_{i \in I}
$$

as in def. \ref{TopologicalVectorBundle} constitute a _[[local trivialization]]_ of $E$.

=--

+-- {: .num_example #VectorBundleSections}
###### Example
**([[section]] of a [[topological vector bundle]])**

Let $E \overset{\pi}{\to} X$ be a topological vector bundle (def. \ref{TopologicalVectorBundle}).

Then a [[homomorphism]] of vector bundles from the [[trivial vector bundle|trivial]] [[line bundle]]
(example \ref{TrivialTopologicalVectorBundle}, remark \ref{TerminologyVectorBundles})

$$
  f \;\colon\; X \times k \longrightarrow E
$$

is, by fiberwise linearity, equivalently a [[continuous function]]

$$
  \sigma \;\colon\;  X \longrightarrow E
$$

such that $\pi \circ \sigma = id_X$;

$$
  f(x, c) = c \sigma(x)
$$

Such functions $\sigma \colon X \to E$ are called _[[sections]]_ (or _cross-sections_)
of the vector bundle $E$.

=--

+-- {: .num_example #TopologicalVetorSubbundle}
###### Example
**(topological vector sub-bundle)**

Given a topological vector bundel $E \to X$ (def. \ref{TopologicalVectorBundle}), then a _sub-bundle_ is a homomorphism of topological vector bundles over $X$

$$
  i\;\colon\; E' \hookrightarrow E
$$

such that for each point $x \in X$ this is a linear embedding of fibers

$$
  i|_x \;\colon\; E'_x \hookrightarrow E_x
  \,.
$$

(This is a [[monomorphism]] in the [[category]] $Vect(X)$ of topological vector bundles over $X$.)

=--

### Transition functions and Cech cohomology
 {#TransitionFunctionsAndCechCohomology}

We discuss how topological vector bundles are equivalently given by [[cocycles]] in [[Cech cohomology]]
constituted by their [[transition functions]].

+-- {: .num_defn #ContinuousFunctionWithValuesInGLn}
###### Definition
**([[continuous functions]] on [[open subsets]] with values in the [[general linear group]])**

For $n \in \mathbb{N}$, regard the [[general linear group]] $GL(n,k)$
as a [[topological group]] with its standard [[topological space|topology]],
given as the [[Euclidean space|Euclidean]] [[subspace topology]]
via $GL(n,k) \subset Mat_{n \times n}(k) \simeq k^{(n^2)}$ or as the
or as the subspace topology $GL(n,k) \subset Maps(k^n, k^n)$ of the [[compact-open topology]] on the [[mapping space]].
(That these topologies coincide is the statement of [this prop.](general+linear+group#AsSubspaceOfTheMappingSpace).

For $X$ a [[topological space]], we write

$$
  \underline{GL(n,k)}
    \;\colon\;
  U \mapsto Hom_{Top}(U, GL(n,k) )
$$

for the assignment that sends an [[open subset]] $U \subset X$ to the [[set]] of [[continuous functions]]
$g \colon U \to GL(n,k)$ (for $U \subset X$ equipped with its [[subspace topology]]), regarded as a
[[group]] via the pointwise group operation in $GL(n,k)$:

$$
   g_1 \cdot g_2 \;\colon\; x \mapsto g_1(x) \cdot g_2(x)
   \,.
$$

Moreover, for $U' \subset U \subset X$ an inclusion of open subsets, and for
$g \in \underline{GL(n,k)}(U)$, we write

$$
  g|_{U'} \in \underline{GL(n,k)}(U')
$$

for the restriction of the continuous function from $U$ to $U'$.

=--

+-- {: .num_remark}
###### Remark
**([[sheaf]] of [[groups]])**

In the language of [[category theory]] the assignment $\underline{GL(n,k)}$
from def. \ref{ContinuousFunctionWithValuesInGLn} of continuous functions to open subsets
and the restriction operations between these is called a _[[sheaf]] of groups
on the [[site of open subsets]]_ of $X$.

=--

+-- {: .num_defn #TransitionFunctions}
###### Definition
**([[transition functions]])

Given a topological vector bundle $E \to X$ as in def. \ref{TopologicalVectorBundle} and a choice of [[local trivialization]]
$\{\phi_i \colon U_i \times k^n \overset{\simeq}{\to} E|_{U_i}\}$ (example \ref{TrivialTopologicalVectorBundle})
there are for $i,j \in I$ induced [[continuous functions]]

$$
  \left\{
    g_{i j} \;\colon\; (U_i \cap U_j) \longrightarrow GL(n, k)
  \right\}_{i,j \in I}
$$

to the [[general linear group]] (as in def. \ref{ContinuousFunctionWithValuesInGLn})
given by composing the local trivialization isomorphisms:

$$
  \array{
    (U_i \cap U_j) \times k^n
      &\overset{ \phi_i|_{U_i \cap U_j} }{\longrightarrow}&
    E|_{U_i \cap U_j}
      &\overset{ \phi_j^{-1}\vert_{U_i \cap U_j} }{\longrightarrow}&
    (U_i \cap U_j) \times k^n
    \\
    (x,v) && \overset{\phantom{AAA}}{\mapsto} && \left( x, g_{i j}(x)(v) \right)
  }
  \,.
$$

These are called the _[[transition functions]]_ for the given local trivialization.

=--

These functions satisfy a special property:

+-- {: .num_defn #CocycleCech}
###### Definition
**([[Cech cohomology|Cech]] [[cocycles]])**

Let $X$ be a [[topological space]].

A _normalized [[Cech cohomology|Cech cocycle]] of degree 1 with [[coefficients]]_
in $\underline{GL(n,k)}$ (def. \ref{ContinuousFunctionWithValuesInGLn})  is

1. an [[open cover]] $\{U_i \subset X\}_{i \in I}$

1. for all $i,j \in I$ a continuous function $g_{i j} \colon U_i \cap U_j \to GL(n,k)$ as in def. \ref{ContinuousFunctionWithValuesInGLn}

such that

1. (normalization) $\underset{i \in I}{\forall}\left( g_{i i} = const_1  \right)  $ (the [[constant function]] on the [[neutral element]] in $GL(n,k)$),

1. (cocycle condition) $\underset{i,j \in I}{\forall}\left(  g_{j k} \cdot g_{i j}  = g_{i k}\;\;\text{on}\, U_i \cap U_j \cap U_k\right)$.

Write

$$
  C^1(X, \underline{GL(n,k)}  )
$$

for the set of all such cocycles for given $n \in \mathbb{N}$ and write

$$
  C^1( X, \underline{GL}(k) )
    \;\coloneqq\;
  \underset{n \in \mathbb{N}}{\sqcup}
   C^1(X, \underline{GL(n,k)})
$$

for the [[disjoint union]] of all these cocycles as $n$ varies.

=--

+-- {: .num_example #CocycleCechTransitionFunction}
###### Example
**([[transition functions]] are [[Cech cohomology|Cech]] [[cocycles]])**

Let $E \to X$ be a topological vector bundle (def. \ref{TopologicalVectorBundle}) and let $\{U_i \subset X\}_{i \in I}$, $\{\phi_i \colon U_i \times k^n \overset{\simeq}{\to} E|_{U_{i}}\}_{i \in I}$ be a local trivialization (example \ref{TrivialTopologicalVectorBundle}).

Then the set of induced [[transition functions]] $\{g_{i j} \colon U_i \cap U_j \to GL(n)\}$ according to def. \ref{TransitionFunctions}
is a _normalized Cech cocycle on $X$ with coefficients in $\underline{GL(k)}$_, according to def. \ref{CocycleCech}.

=--

+-- {: .proof}
###### Proof

This is immediate from the definition:

$$
  \begin{aligned}
    g_{i i }(x)
      & =
      \phi_i^{-1} \circ \phi_i(x,-)
      \\
      & = id_{k^n}
  \end{aligned}
$$

and

$$
  \begin{aligned}
    g_{j k}(x) \cdot g_{i j}(x)
      & =
    \left(\phi_k^{-1} \circ \phi_j\right) \circ \left(\phi_j^{-1}\circ \phi_i\right)(x,-)
    \\
      & =
    \phi_k^{-1} \circ \phi_i(x,-)
    \\
    & =
    g_{i k}(x)
  \end{aligned}
  \,.
$$

=--

Conversely:

+-- {: .num_example #TopologicalVectorBundleFromCechCocycle}
###### Example
**(topological vector bundle constructed from a [[Cech cohomology|Cech]] [[cocycle]])

Let $X$ be a [[topological space]]
and let $c \in C^1(X, \underline{GL(k)})$ a Cech cocycle on $X$ according to def. \ref{CocycleCech},
with open cover $\{U_i \subset X\}_{i \in I}$ and component functions $\{g_{i j}\}_{i,j \in I}$.

This induces an [[equivalence relation]] on the [[product topological space]]

$$
  \left(
    \underset{i \in I}{\sqcup} U_i
  \right)
  \times
  k^n
$$

(of the [[disjoint union space]] of the patches $U_i \subset X$ regarded as [[topological subspaces]]
with the [[product space]] $k^n = \underset{\{1,\cdots, n\}}{\prod} k$) given by

$$
  \big(
    ((x,i), v)
     \;\sim\;
    ((y,j), w)
  \big)
   \;\Leftrightarrow\;
  \left(
     (x = y)
       \;\text{and}\;
     (g_{i j}(x)(v) = w)
  \right)
  \,.
$$

Write

$$
  E(c)
   \;\coloneqq\;
  \left(
    \left(
      \underset{i \in I}{\sqcup} U_i
    \right)
      \times
    k^n
  \right)
  /
  \left(
     \left\{
        g_{i j}
     \right\}_{i,j \in I}
  \right)
$$

for the resulting [[quotient topological space]]. This comes with the evident projection

$$
  \array{
    E(c) &\overset{\phantom{AA}\pi \phantom{AA}}{\longrightarrow}& X
    \\
    [(x,i,),v] &\overset{\phantom{AAA}}{\mapsto}& x
  }
$$

which is a [[continuous function]] (by the [[universal property]] of the [[quotient topological space]] construction, since the corresponding continuous [[function]] on the un-quotientd disjoint union space respects the equivalence relation). Moreover, each [[fiber]] of this map is identified with $k^n$, and hence canonicaly carries the structure of a [[vector space]].

Finally, the quotient co-projections  constitute a local trivialization of this vector bundle over the given open cover.

Therefore $E(c) \to X$ is a topological vector bundle (def. \ref{TopologicalVectorBundle}). We say it is the topological vector bundle _glued from the transition functions_.

=--

+-- {: .num_remark}
###### Remark
**(bundle glued from [[Cech cohomology|Cech]] [[cocycle]] is a [[coequalizer]])**

Stated more [[category theory|category theoretically]], the constructure of a topological vector bundle from Cech cocycle data in example \ref{TopologicalVectorBundleFromCechCocycle} is a [universal construction in topological spaces](Top#UniversalConstructions), namely the [[coequalizer]] of the two morphisms

$$i, \mu: \underset{i j}{\sqcup} (U_i \cap U_j) \times V \overset{\to}{\to} \underset{i}{\sqcup} U_i \times V$$

in the category of vector space objects in the slice category $Top/X$. Here the restriction of $i$ to the coproduct summands is induced by inclusion:

$$(U_i \cap U_j) \times V \hookrightarrow U_i \times V \hookrightarrow \underset{i}{\sqcup} U_i \times V$$

and the restriction of $\mu$ to the coproduct summands is
via the action of the transition functions:

$$(U_i \cap U_j) \times V \overset{(\langle incl, g_{i j} \rangle) \times V}{\to} U_j \times GL(V) \times V \overset{action}{\to} U_j \times V \hookrightarrow \underset{j}{\sqcup} U_j \times V$$

=--

In fact, extracting transition functions from a vector bundle by def. \ref{TransitionFunctions} and constructing a vector bundle from Cech coycle data as above are operations that are inverse to each other, up to [[isomorphism]].

+-- {: .num_prop #FromTransitionFunctionsReconstructVectorBundle}
###### Proposition
**(topological vector bundle reconstructed from its [[transition functions]])**

Let $[E \overset{\pi}{\to} X]$ be a [[topological vector bundle]] (def. \ref{TopologicalVectorBundle}),
let $\{U_i \subset X\}_{i \in I}$ be an [[open cover]] of the base space, and
let $\left\{  U_i \times k^n \underoverset{\simeq}{\phi_i}{\longrightarrow} E|_{U_i}  \right\}_{i \in I}$
be a [[local trivialization]].

Write

$$
  \left\{ g_{i j} \coloneqq  \phi_j^{-1}\circ \phi_i  \colon U_i \cap U_j \to GL(n,k) \right\}_{i,j \in I}
$$

for the corresponding [[transition functions]] (def. \ref{TransitionFunctions}). Then there is an
[[isomorphism]] of vector bundles over $X$

$$
  \left(
    \left(
      \underset{i \in I}{\sqcup} U_i
    \right)
      \times
    k^n
  \right)
  /
  \left(
     \left\{
        g_{i j}
     \right\}_{i,j \in I}
  \right)
    \;\underoverset{\simeq}{(\phi_i)_{i \in I}}{\longrightarrow}\;
  E
$$

from the vector bundle glued from the transition functions according to def. \ref{TransitionFunctions} to the original bundle $E$,
whose components are the original local trivialization isomorphisms.

=--

+-- {: .proof}
###### Proof

By the [[universal property]] of the [[disjoint union space]] ([[coproduct]] in [[Top]]),
continuous functions out of them are equivalently sets of continuous functions out of every
summand space. Hence the set of local trivializations $\{U_i \times k^n \underoverset{\simeq}{\phi_i}{\to} E|_{U_i} \subset E\}_{i \in I}$
may be collected into a single [[continuous function]]

$$
    \underset{i \in I}{\sqcup} U_i
     \times
   k^n
   \overset{(\phi_i)_{i \in I}}{\longrightarrow }
   E
   \,.
$$

By construction this function respects the [[equivalence relation]] on the disjoint union space given by
the transition functions, in that for each $x \in U_i \cap U_j$ we have

$$
  \phi_i((x,i),v)
  =
  \phi_j \circ \phi_j^{-1} \circ \phi_i((x,i),v)
  =
  \phi_j \circ ((x,j),g_{i j}(x)(v))
  \,.
$$

By the [[universal property]] of the [[quotient space]] coprojection this means that $(\phi_i)_{i \in I}$
uniquely [[extension|extends]] to a [[continuous function]] on the quotient space such that the
following [[commuting diagram|diagram commutes]]

$$
  \array{
    \left(
      \underset{i \in I}{\sqcup} U_i
    \right)
      \times
    k^n
    &\overset{(\phi_i)_{i \in I}}{\longrightarrow}&
  E
  \\
  \downarrow  & \nearrow_{\exists !}
  \\
  \left(
    \left(
      \underset{i \in I}{\sqcup} U_i
    \right)
      \times
    k^n
  \right)
  /
  \left(
     \left\{
        g_{i j}
     \right\}_{i,j \in I}
  \right)
  }
  \,.
$$

It is clear that this continuous function is a [[bijection]]. Hence to show that it is a
[[homeomorphism]], it is now sufficient to show that this is an [[open map]] (by [this prop.](Introduction+to+Topology+--+1#HomeoContinuousOpenBijection)).

So let $O$  be an subset in the quotient space which is open. By definition of the
[[quotient topology]] this means equivalently that its restriction $O_i$ to $U_i \times k^n$
is open for each $i \in I$. Since the $\phi_i$ are homeomorphsms, it follows that the
images $\phi_i(O_i)  \subset  E\vert_{U_ i}$ are open.
By the nature of the [[subspace topology]], this means that these images are open also in $E$.
Therefore also the union $f(O) = \underset{i \in I}{\cup} \phi_i(O_i)$ is open.

=--

+-- {: .num_defn #CoboundaryCech}
###### Definition
**([[coboundary]] between [[Cech cohomology|Cech]] [[cocycles]] )

Let $X$ be a [[topological space]] and let $c_1, c_2 \in C^1(X, \underline{GL(k)})$ be two [[Cech cohomology|Cech]] [[cocycles]]
(def. \ref{CocycleCech}), given by

1. $\{U_i \subset X\}_{i \in I}$ and $\{U'_i \subset X\}_{i' \in I'}$ two [[open covers]],

1. $\{g_{i j} \colon U_i \cap U_j \to GL(k,n_)\}_{i,j \in I}$ and $\{g_'_{i',j'} \colon U'_{i'} \cap U'_{j'} \to GL(n',k) \}_{i', j'}$  the corrsponding component functions.

Then a _[[coboundary]]_  between these two cocycles is

1. the condition that $n = n'$,

1. an [[open cover]] $\{V_\alpha \subset X\}_{\alpha \in A}$,

1. [[functions]] $\phi \colon A \to I$ and $\phi' \colon A \to J$ such that $\underset{\alpha \in A}{\forall}\left( \left( V_\alpha \subset U_{\phi(\alpha)} \right) \,\text{and}\, \left( V_\alpha \subset U'_{\phi'(\alpha)} \right) \right)$

1. a set $\{ \kappa_\alpha \colon V_\alpha \to  GL(n,k) \}$ of continuous functions as in def. \ref{CocycleCech}

such that

* $\underset{ \alpha, \beta \in A }{\forall} \left(  \kappa_{\beta} \cdot g_{\phi(\alpha) \phi(\beta)}  = g'_{\phi'(\alpha) \phi'(\beta)} \cdot \kappa_{\alpha} \,\, \text{on}\,\, V_\alpha \cap V_\beta \right) $,

  hence such that the following [[diagrams]] of [[linear maps]] [[commuting diagram|commute]] for all $\alpha, \beta \in A$ and $x \in V_{\alpha} \cap V_\beta$:

  $$
    \array{
      k^n  &\overset{ g_{\phi(\alpha) \phi(\beta)}(x) }{\longrightarrow}& k^n
      \\
      {}^{\mathllap{\kappa_{\alpha}(x)} }\downarrow && \downarrow^{\mathrlap{ \kappa_{\beta}(x) }}
      \\
      k^n  &\underset{ g'_{\phi'(\alpha) \phi'(\beta)}(x) }{\longrightarrow}& k^n
    }
    \,.
  $$

Say that two Cech cocycles are _cohomologous_ if there exists a coboundary between them.

=--

+-- {: .num_example #FinerCoverCech}
###### Example
**([[refinement]] of a [[Cech  cohomology|Cech]] [[cocycle]] is a [[coboundary]])**

Let $X$ be a [[topological space]] and let $c \in C^1(X, \underline{GL(k)})$ be a Cech cocycle as in def. \ref{CocycleCech},
with respect to some open cover $\{U_i \subset X\}_{i \in I}$ given by component functions $\{g_{i j}\}_{i,j \in I}$.

Then for $\{V_\alpha \subset X\}_{\alpha \in A}$ a [[refinement]] of the given open cover, hence an open cover such that
there exists a [[function]] $\phi \colon A \to I$ with $\underset{\alpha \in A}{\forall}\left( V\alpha \subset U_{\phi(\alpha)}  \right)$,
then

$$
  g'_{ \alpha \beta }
  \coloneqq g_{\phi(\alpha) \phi(\beta)}
  \colon V_\alpha \cap V_\beta
    \longrightarrow
  GL(n,k)
$$

are the components of a Cech cocycle $c'$ which is cohomologous to $c$.

=--

+-- {: .num_prop #CechCoboundaryFromIsomorphismBetweenVectoreBundles}
###### Proposition
**([[isomorphism]] of topological vector bundles induces [[Cech cohomology|Cech]] [[coboundary]] between their [[transition functions]])**

Let $X$ be a topological space, and let $c_1, c_2 \in C^1(X, \underline{GL(n,k)}  )$ be two Cech cocycles as in def. \ref{CocycleCech}.

Every [[isomorphism]] of topological vector bundles

$$
  f
    \;\colon\;
  E(c_1)
    \overset{\simeq}{\longrightarrow}
  E(c_2)
$$

between the vector bundles glued from these cocycles according to def. \ref{TopologicalVectorBundleFromCechCocycle}
induces a coboundary between the two cocycles,

$$
  c_1 \sim c_2
  \,,
$$

according to def. \ref{CoboundaryCech}.

=--

+-- {: .proof}
###### Proof

By example \ref{FinerCoverCech} we may assume without restriction that the two Cech cocycles are
defined with respect to the same open cover $\{U_i \subset X\}_{i \in I}$ (for if they are not, then both are cohomologous to
cocycles on a joint refinement of the original covers and we may argue with these).

Accordingly, by example \ref{TopologicalVectorBundleFromCechCocycle} the two bundles $E(c_1)$ and $E(c_2)$
both have local trivializations of the form

$$
  \{ U_i \times k^n \underoverset{\simeq}{\phi^1_i}{\longrightarrow} E(c_1)\vert_{U_i}\}
$$

and

$$
  \{ U_i \times k^n \underoverset{\simeq}{\phi^2_i}{\longrightarrow} E(c_2)\vert_{U_i}\}
$$

over this cover.  Consider then for $i \in I$ the function

$$
  f_i \coloneqq  (\phi_i^2)^{-1}\circ f\vert_{U_i} \circ  \phi^1_i
  \,,
$$

hence the unique function making the following [[commuting diagram|diagram commute]]:

$$
  \array{
    U_i \times k^n &\underoverset{\simeq}{\phi^1_i}{\longrightarrow}& E(c_1)\vert_{U_i}
    \\
    {}^{\mathllap{f_i}}\downarrow && \downarrow^{\mathrlap{ f }}
    \\
    U_i \times k^n &\underoverset{\phi^2_i}{\simeq}{\longrightarrow}& E(c_2)\vert_{U_i}
  }
  \,.
$$

This induces for all $i,j \in I$ the following composite commuting diagram

$$
  \array{
    (U_i \cap U_j) \times k^n
      &\underoverset{\simeq}{\phi^1_i}{\longrightarrow}&
    E(c_1)\vert_{U_i \cap U_j}
      & \underoverset{\simeq}{(\phi^1_j)^{-1}}{\longrightarrow}  &
    (U_i \cap U_j) \times k^n
    \\
    {}^{\mathllap{f_i}}\downarrow
      &&
    \downarrow^{\mathrlap{ f }}
      &&
    \downarrow^{\mathrlap{ f_j }}
    \\
    (U_i \cap U_j) \times k^n
      &\underoverset{\phi^2_i}{\simeq}{\longrightarrow}&
    E(c_2)\vert_{U_1 \cap U_2}
      &\underoverset{(\phi^2_j)^{-1}}{\simeq}{\longrightarrow}&
    (U_i \cap U_j) \times k^n
  }
  \,.
$$

By construction, the two horizonal composites of this diagram are pointwise
given by the components $g^1_{i j}$ and $g^2_{i j}$of the cocycles $c_1$ and $c_2$, respectively.
Hence the commutativity of this
diagram is equivalently the commutativity of these diagrams:

  $$
    \array{
      k^n  &\overset{ g^1_{i j}(x) }{\longrightarrow}& k^n
      \\
      {}^{\mathllap{ f_i(x) } }\downarrow && \downarrow^{\mathrlap{ f_j(x) }}
      \\
      k^n  &\underset{ g^2_{ i j }(x) }{\longrightarrow}& k^n
    }
    \,.
  $$

for all $i,j \in I$ and $x  \in U_i \cap U_j$. By def. \ref{CoboundaryCech} this exhibits the required coboundary.

=--

+-- {: .num_defn #CohomologyCech}
###### Definition
**([[Cech cohomology]])**

Let $X$ be a [[topological space]]. The relation $\sim$ on [[Cech cohomology|Cech cocycles]]
of being cohomologous (def. \ref{CoboundaryCech})
is an [[equivalence relation]] on the set $C^1( X, \underline{GL(k)} )$ of [[Cech cohomology|Cech]] [[cocycles]] (def. \ref{CocycleCech}).

Write

$$
  H^1(X, \underline{GL(k)} )
   \;\coloneqq\;
  C^1(X, \underline{GL(k)} )/\sim
$$

for the resulting set of [[equivalence classes]]. This is called the
_[[Cech cohomology]] of $X$ in degree 1 with [[coefficients]] in $\underline{GL(k)}$.

=--

+-- {: .num_prop}
###### Proposition
**(degree-1 [[Cech cohomology]] computes [[topological vector bundles]])

Let $X$ be a [[topological space]].

The construction of gluing a topological vector bundle from a Cech cocycle (example \ref{TopologicalVectorBundleFromCechCocycle})
constitutes a [[natural bijection|bijection]]
between the degree-1 Cech cohomology of $X$ with coefficients in $GL(n,k)$ (def. \ref{CohomologyCech})
and the set of [[isomorphism classes]] of topological vector bundles on $X$ (def. \ref{TopologicalVectorBundle}, remark \ref{TopologicalVectorBundlesCategory}):

$$
  \array{
    H^1(X,\underline{GL(k)})
      &\overset{\phantom{AA}\simeq \phantom{AA}}{\longrightarrow}&
    Vect(X)_{/\sim}
    \\
    c &\overset{\phantom{AAA}}{\mapsto}& E(c)
  }
  \,.
$$

=--

+-- {: .proof}
###### Proof

First we need to see that the function is well defined, hence that if
cocycles $c_1, c_2 \in C^1(X,\underline{GL(k)})$ are
related by a coboundary, $c_1 \sim c_2$ (def. \ref{CoboundaryCech}),
then the vector bundles $E(c_1)$ and $E(c_2)$ are related by an isomorphism.

Let $\{V_\alpha \subset X\}_{\alpha \in A}$ be the open cover with respect to which the coboundary
$\{\kappa_\alpha \colon V_\alpha \to GL(n,k)\}_{\alpha}$ is defined, with refining functions
$\phi \colon A \to I$ and $\phi' \colon A \to I'$.  Let
$\left\{ \mathbb{R}^n \underoverset{\simeq}{\psi_{\phi(\alpha)}\vert_{V_\alpha}  }{\to} E(c_1)\vert_{V_\alpha} \right\}_{\alpha \in A}$
and $\left\{ V_\alpha \times k^n \underoverset{\simeq}{\psi'_{\phi'(\alpha)}\vert_{V_\alpha}  }{\to} E(c_2)\vert_{V_\alpha} \right\}_{\alpha \in A}$
be the corresponding restrictions of the canonical local trivilizations of the two glued bundles.

For $\alpha \in A$ define

$$
  f_\alpha \coloneqq \psi'_{\phi'(\alpha)}\vert_{V_\alpha}  \circ \kappa_\alpha \circ (\psi_{\phi(\alpha)}\vert_{V_\alpha}  )^{-1}
  \phantom{AAAA}
    \text{hence:}
  \phantom{AAA}
  \array{
    V_\alpha \times k^n
     &\overset{ \psi_{\phi(\alpha)}\vert_{V_\alpha} }{\longrightarrow}&
    E(c_1)\vert_{V_\alpha}
    \\
    {}^{\mathllap{\kappa_\alpha}}\downarrow && \downarrow^{\mathrlap{f_\alpha}}
    \\
    V_\alpha \times k^n
     &\overset{ (\psi'_{\phi'(\alpha)}\vert_{V_\alpha})^{-1} }{\longleftarrow}&
    E(c_1)\vert_{V_\alpha}
  }
  \,.
$$

Observe that for $\alpha, \beta \in A$ and $x \in V_\alpha \cap V_\beta$ the coboundary condition
implies that

$$
  f_\alpha\vert_{V_\alpha \cap V_\beta}
  \;=\;
  f_\beta\vert_{V_\alpha \cap V_\beta}
$$

because in the diagram

$$
  \array{
    k^n &\overset{ g_{\phi(\alpha) \phi(\beta) }(x) }{\longrightarrow}& k^n
    \\
    {}^{\mathllap{\kappa_\alpha(x)}}\downarrow
      &&
    \downarrow^{\mathrlap{\kappa_{\beta}(x)}}
    \\
    k^n &\underset{g'_{\phi'(\alpha) \phi'(\beta)}(x)  }{\longrightarrow}& k^n
  }
  \phantom{AAAAA}
    =
  \phantom{AAAAA}
  \array{
    k^n
      &\overset{ \psi_{\phi(\alpha)}(x) }{\longrightarrow}&
    E(c_1)_x
      &\overset{ (\psi_{\phi(\beta)})^{-1}(x) }{\longrightarrow}&
    k^n
     \\
    {}^{\mathllap{\kappa_\alpha(x)}}\downarrow
      &&
    \downarrow^{\mathrlap{\exists !} }
      &&
    \downarrow^{\mathrlap{\beta_\alpha(x)}}
    \\
    k^n
      &\overset{ \psi'_{\phi'(\alpha)}(x) }{\longrightarrow}&
    E(c_2)_x
      &\overset{ (\psi'_{\phi'(\beta)})^{-1}(x) }{\longrightarrow}&
    k^n
 }
$$

the vertical morphism in the middle on the right is unique, by the fact that all other
morphisms in the diagram on the right are invertible.

Therefore there is a unique vector bundle homomorphism

$$
  f\;\colon\; E(c_1) \to E(c_2)
$$

given for all $\alpha \in A$ by $f\vert_{V_\alpha} = f_\alpha$. Similarly there is a unique vector bundle
homomorphism

$$
  f^{-1}\;\colon\; E(c_2) \to E(c_1)
$$

given for all $\alpha \in A$ by $f^{-1}\vert_{V_\alpha} = f^{-1}_\alpha$.
Hence this is the required vector bundle isomorphism.

Finally to see that the function from Cech cohomology classes to isomorphism classes of vector
bundles thus defined is a bijection:

By prop. \ref{FromTransitionFunctionsReconstructVectorBundle} the function is [[surjective function|surjective]],
and by prop. \ref{CechCoboundaryFromIsomorphismBetweenVectoreBundles} it is injective.

=--

$\,$

## Examples
 {#Examples}

+-- {: .num_example #TautologicalLineBundle}
###### Example
**([[tautological line bundle]])**

For $n \in \mathbb{N}$ then the [[projective space]] $k P^n$
carries the _[[tautological line bundle]]_ whose fiber over the $k$-line $[v] \in k P^n$
is that $k$-line.

For details see [there](tautological+line+bundle#AsAtopologicalLieBundle)

=--

+-- {: .num_example}
###### Example
**([[cylinder]])**

Let

$$
  S^1
    =
  \left\{
    (x,y)
     \;\vert\;
     x^2 + y^2 = 1
  \right\}
  \;\subset\,
  \mathbb{R}^2
$$

<div style="float:right;margin:0 10px 10px 0;">
<img src="https://ncatlab.org/nlab/files/cylinder.jpg" width="190">
</div>

be the [[circle]] with its  [[Euclidean space|Euclidean]] [[subspace]] [[metric topology]].

Then the [[trivial vector bundle|trivial]] [[real vector bundle|real]] [[line bundle]] on the circle is the the [[cylinder]]

$$
  S^1 \times \mathbb{R}
$$

=--

+-- {: .num_example #MoebiusStrip}
###### Example
**([[Moebius strip]])**

Let

$$
  S^1
    =
  \left\{
    (x,y)
     \;\vert\;
     x^2 + y^2 = 1
  \right\}
  \;\subset\,
  \mathbb{R}^2
$$

be the  [[circle]] with its  [[Euclidean space|Euclidean]] [[subspace]] [[metric topology]].
Consider the [[open cover]]

$$
  \left\{
    U_n \subset S^1
  \right\}_{n \in \{0,1,2\}}
$$

with

$$
  U_n \coloneqq \left\{ (cos(\alpha), sin(\beta)) \;\vert\; n \frac{2 \pi }{3} - \epsilon \lt \alpha \lt (n+1) \frac{2\pi }{3} + \epsilon  \right\}
$$

for any $\epsilon \in (0,2\pi/6)$.

Define a [[Cech cohomology]] cocycle (remark \ref{CechCoycleCondition}) on this cover by

$$
  g_{n_1 n_2} =
  \left\{
    \array{
      const_{-1} & \vert & (n_1,n_2) = (0,2)
      \\
      const_{-1} &\vert& (n_1,n_2) = (2,0)
      \\
      const_1 &\vert& \text{otherwise}
    }
  \right.
$$

<div style="float:right;margin:0 10px 10px 0;">
<img src="https://ncatlab.org/nlab/files/moebiusstrip.jpg" width="200">
</div>

Since there are no non-trivial triple intersections, all cocycle conditions are
evidently satisfied.

Accordingly by example \ref{TopologicalVectorBundleFromCechCocycle} these functions define a vector bundle.
This is the _[[Moebius strip]]_

=--

+-- {: .num_example }
###### Example
**([[basic complex line bundle on the 2-sphere]])**

Let

$$
  S^2
    \coloneqq
  \left\{
    (x,y,z)
    \;\vert\;
    x^2 + y^2 + z^2 = 1
  \right\}
    \subset
  \mathbb{R}^3
$$

be the [[2-sphere]] with its [[Euclidean space|Euclidean]] [[subspace]] [[metric topology]].
Let

$$
  \left\{
    U_{i} \subset S^2
  \right\}_{i \in \{+,-\}}
$$

be the two [[complements]] of antipodal points

$$
  U_\pm \coloneqq S^2 \setminus \{(0, 0, \pm 1)\}
  \,.
$$

Define continuous functions

$$
  \array{
    U_+ \cap U_-
      &\overset{g_{\pm \mp}}{\longrightarrow}&
    GL(1,\mathbb{C})
    \\
    ( \sqrt{1-z^2} \, cos(\alpha), \sqrt{1-z^2} \, sin(\alpha), z)
      &\mapsto&
    \exp(\pm 2\pi i \alpha)
  }
  \,.
$$

Since there are no non-trivial triple intersections, the only cocycle condition is

$$
  g_{\mp \pm} g_{\pm \mp} = g_{\pm \pm} = id
$$

which is clearly satisfied.

The [[complex line bundle]] this defined is called the _[[basic complex line bundle on the 2-sphere]]_.

With the 2-sphere identified with the [[complex projective space]] $\mathbb{C} P^1$
(the [[Riemann sphere]]),
the basic complex line bundle is the [[tautological line bundle]] (example \ref{TautologicalLineBundle}) on $\mathbb{C}P^1$.

=--

+-- {: .num_example #ClutchingConstruction}
###### Example
**([[clutching construction]])**

Generally, for $n \in \mathbb{N}$, $n \geq 1$ then the [[n-sphere]]
$S^n$ may be covered by two open [[hemispheres]] intersecting in an
[[equator]] of the form $S^{n-1} \times (-\epsilon, \epsilon)$.
A vector bundle is then defined by specifying a single function

$$
  g_{+-}
  \;\colon\;
  S^{n-1} \longrightarrow GL(n,k)
  \,.
$$

This is called the _[[clutching construction]]_ of vector bundles over [[n-spheres]].

=--

+-- {: .num_example}
###### Example
**([[tangent bundle]])**

For $X$ the [[topological space]] underlyithe ng a [[differentiable manifold]] then its [[tangent bundle]]
$T X$ is a [[real vector bundle]] over $X$ whose [[rank of a vector bundle|rank]] is the [[dimension]] of $X$.

=--

+-- {: .num_example}
###### Example
**([[normal bundle]])**

For $i X \hookrightarrow Y$ an [[embedding of differentiable manifolds]], then
the _[[normal bundle]]_

$$
  N_i X \coloneqq T Y/T X
$$

is the real vector bundle over $Y$ whose [[fiber]] at $x \in X$ is the [[quotient vector space]]
$(N_i X)_x \coloneqq T_{i(x)} Y / T_x X$.

=--

## Properties

### Basic properties
 {#BasicProperties}

+-- {: .num_lemma #FiberwiseIsoisIsomorphismOfVectorBundles}
###### Lemma
**(homomorphism of vector bundles is isomorphism as soon as it is a fiberwise isomorphism)**

Let $[E_1 \to X]$ and $[E_2 \to X]$ be two topological vector bundles (def. \ref{TopologicalVectorBundle}).

If a [[homomorphism]] of vector bundles $f \colon E_1 \longrightarrow E_2$
restricts on the [[fiber]] over each point to a linear isomorphism

$$
  f\vert_x \;\colon\; (E_1)_x \overset{\simeq}{\longrightarrow} (E_2)_x
$$

then $f$ is already an isomorphism of vector bundles.

=--

+-- {: .proof}
###### Proof

It is clear that $f$ has an [[inverse function]]
of underlying sets $f^{-1} \colon E_2 \to _E_1$ which is a function over $X$: Over each $x \in X$
it it the linear inverse $(f\vert_x)^{-1} \colon (E_2)_x \to (E_1)_x$.

What we need to show is that this is a continuous function.

By remark \ref{CommonOpenCoverLocalTrivialization} we find an open cover $\{U_i \subset X\}_{i \in I}$ over which
both bundles have a local trivialization.

$$
  \left\{ U_i \underoverset{\simeq}{\phi^1_i}{\to} (E_1)\vert_{U_i}\right\}_{i \in I}
  \phantom{AA}
  \text{and}
  \phantom{AA}
  \left\{ U_i \underoverset{\simeq}{\phi^2_i}{\to} (E_2)\vert_{U_i} \right\}_{i \in I}
  \,.
$$

Restricted to any patch $i \in I$ of this cover, the homomorphism $f|_{U_i}$ induces a homomorphism
of  [[trivial vector bundles]]

$$
  f_i \coloneqq \phi^2_j^{-1} \circ f \circ \phi^1_i
   \phantom{AAAAAA}
  \array{
    U_i \times k^n &\underoverset{\simeq}{\phi^1_i}{\longrightarrow}& (E_1)\vert|_{U_i}
    \\
    {}^{f_i}\downarrow && \downarrow^{\mathrlap{f\vert_{U_i}}}
    \\
    U_i \times k^n &\underoverset{\phi^2_i}{\simeq}{\longrightarrow}& (E_2)\vert_{U_j}
  }
  \,.
$$

Also the $f_i$ are fiberwise invertible, hence are
continuous bijections. We claim that these are [[homeomorphisms]], hence that their
inverse functions $(f_i)^{-1}$ are also continuous.

To this end we re-write the $f_i$ a little. First observe that by the [[universal property]]
of the [[product topological space]] and since they fix the base space $U_i$, the $f_i$ are equivalently given by
a continuous function

$$
   h_i \;\colon\; U_i \times k^n \longrightarrow k^n
$$

as

$$
  f_i(x,v) = (x, h_i(x,v))
  \,.
$$

Moreovern since $k^n$ is [[locally compact topological space|locally compact]] (like every [[finite dimensional vector space]], by the [[Heine-Borel theorem]]),
the [[mapping space]] [[adjunction]] says (by [this prop.](Introduction+to+Topology+--+1#UniversalPropertyOfMappingSpace)) that there is a continuous function

$$
  \tilde h_i
    \;\colon\;
  U_i \longrightarrow Maps(k^n, k^n)
$$

(with $Maps(k^n,k^n)$ the set of continuous functions $k^n \to k^n$ equipped with the [[compact-open topology]])
which factors $h_i$ via the [[evaluation]] map as

$$
  h_i
    \;\colon\;
  U_i \times k^n
    \overset{\tilde h_i \times id_{k^n}}{\longrightarrow}
  Maps(k^n, k^n) \times k^n
    \overset{ev}{\longrightarrow}
  k^n
  \,.
$$

By assumption of fiberwise
linearity the functions $\tilde h_i$ in fact take values in the [[general linear group]]

$$
  GL(n,k) \subset Maps(k^n, k^n)
$$

and this inclusion is a [[homeomorphism]] onto its image (by [this prop.](general+linear+group#AsSubspaceOfTheMappingSpace)).

Since passing to [[inverse matrices]]

$$
  (-)^{-1} \;\colon\; GL(n,k) \longrightarrow GL(n,k)
$$

is a [[rational function]] on its domain $GL(n,k) \subset Mat_{n \times n}(k) \simeq k^{(n^2)}$ inside [[Euclidean space]]
and since [[rational functions are continuous]] on their domain of definition,
it follows that the inverse of $f_i$

$$
  (f_i)^{-1}
    \;\colon\;
  U_i \times k^n
    \overset{(id  , \tilde h_i )  }{\longrightarrow}
  U_i \times k^n \times GL(n,k)
    \overset{ id  \times (-)^{-1} }{\longrightarrow}
  U_i \times k^n \times GL(n,k)
    \overset{id  \times ev}{\longrightarrow}
  U_i \times k^n
$$

is a continuous function.

To conclude that also $f^{-1}$ is a continuous function we make use prop. \ref{FromTransitionFunctionsReconstructVectorBundle}
to find an isomorphism between $E_2$ and a [[quotient topological space]] of the form

$$
  E_2 \simeq \left(\underset{i \in I}{\sqcup} (U_i \times k^n) \right) / \left(  \left\{ g_{i j}\right\}_{i,j\in I} \right)
  \,.
$$

Hence $f^{-1}$ is equivalently a function on this quotient space, and we need to show that as such it is continuous.

By the [[universal property]] of the [[disjoint union space]] (the [[coproduct]] in [[Top]]) the set of continuous functions

$$
  \{ U_i \times k^n \overset{f_i^{-1}}{\to} U_i \times k^n \overset{\phi^1_i}{\to} E_1 \}_{i \in I}
$$

corresponds to a single continuous function of the form

$$
  (\phi^1_i \circ f_i^{-1})_{i \in I}
    \;\colon\;
  \underset{i \in I}{\sqcup} U_i \times k^n
    \longrightarrow
  E_1
  \,.
$$

These functions respect the equivalence relation, since for each $x \in U_i \cap U_j$ we have

$$
  (\phi^1_i \circ f_i^{-1})((x,i),v)
    =
  (\phi^1_j \circ f_j^{-1})( (x,j), g_{i j}(x)(v) )
  \phantom{AAAA}
  \text{since:}
  \phantom{AAAA}
  \array{
    && E_1
    \\
     & {}^{\mathllap{\phi^1_i \circ f_i^{-1}}}\nearrow
     &
       \uparrow^{\mathrlap{f^{-1}}}
     & \nwarrow^{\mathrlap{ \phi^1_j \circ f_j^{-1} }}
    \\
    U_i \times k^n
      &\underset{\phi^2_i}{\longrightarrow}&
    (E_2)\vert_{U_i \cap U_i}
      &\underset{(\phi^2_j)^{-1}}{\longrightarrow}&
    U_i \times k^n
  }
  \,.
$$

Therefore by the [[universal property]] of the [[quotient topological space]] $E_2$, these functions
[[extension|extend]] to a unique continuous function $E_2 \to E_1$ such that the following
[[commuting diagram|diagram commutes]]:

$$
  \array{
    \underset{i \in i}{\sqcup} U_i \times k^n
      &\overset{( \phi^1_i \circ f_i^{-1} )_{i \in I}}{\longrightarrow}&
    E_1
    \\
    \downarrow & \nearrow_{\mathrlap{\exists !}}
    \\
    E_2
  }
  \,.
$$

This unique function is clearly $f^{-1}$ (by pointwise inspection) and therefore $f^{-1}$ is continuous.

=--

+-- {: .num_example #FiberwiseLinearlyIndependentSectionsTrivialize}
###### Example
**(fiberwise linearly independent sections trivialize a vector bundle)**

If a topological vector bundle $E \to X$ of [[rank of a vector bundle|rank]] $n$
admits $n$ [[sections]] (example \ref{VectorBundleSections})

$$
  \{\sigma_k \;\colon\; X \longrightarrow E\}_{k \in \{1, \cdots, n\}}
$$

that are linearly independent at each point $x \in X$, then $E$ is trivializable (example \ref{TrivialTopologicalVectorBundle}).
In fact, with the sections regarded as vector bundle homomorphisms out of the trivial vector bundle
of rank $n$ (according to example \ref{VectorBundleSections}), these sections _are_ the trivialization

$$
  (\sigma_1, \cdots, \sigma_n)
  \;\colon\;
  (X \times k^n)
    \overset{\simeq}{\longrightarrow}
  E
  \,.
$$

This is because their linear independence at each point means precisely that this
morphism of vector bundles is a fiber-wise linear isomorphsm
and therefore an isomorphism of vector bundles by lemma \ref{FiberwiseIsoisIsomorphismOfVectorBundles}.

=--

### Direct summand bundles
 {#DirectSummandBundles}

We discuss properties of the [[direct sum of vector bundles]] for topological vector bundles.

+-- {: .num_prop #TopologicalSubBundlesOverParacompactHausdorffSpacesAreDirectSummands}
###### Proposition
**(sub-bundles over [[paracompact spaces]] are [[direct sum of vector bundles|direct summands]])**

Let

1. $X$ be a [[paracompact Hausdorff space]],

1. $E \to X$ a [[topological vector bundle]] (def. \ref{TopologicalVectorBundle}).

Then every topological vector sub-bundle $E_1 \hookrightarrow E$ (example \ref{TopologicalVetorSubbundle}) is a direct vector bundle summand, in that there exists another vector sub-bundle $E_2 \hookrightarrow E$ (example \ref{TopologicalVetorSubbundle}) such that their [[direct sum of vector bundles]] is $E$:

$$
  E_1 \oplus E_2 \simeq E
  \,.
$$

=--

([e.g. Hatcher, prop. 1.3](#Hatcher))

+-- {: .proof}
###### Proof

Since $X$ is assumed to be paracompact Hausdorff, there exists an [[inner product on vector bundles]]

$$
  \langle -,-\rangle
  \;\colon\;
  E \oplus_X E
    \longrightarrow
  X \times k
$$

(by [this prop.](inner+product+of+vector+bundles#ExistenceOfInnerProductOfTopologicalVectorBundlesOverParacompactHausdorffSpaces)). This defines at each $x \in X$ the [[orthogonal complement]] $(E'_x)^\perp \subset E_x$ of $E'_x \hookrightarrow E$. The [[subspace]] of these orthogonal complements is readily checked to be a [[topological vector bundle]] $(E')^\perp \to X$. Hence by construction we have

$$
  E \;\simeq\; E' \oplus_X (E')^\perp
  \,.
$$

=--

+-- {: .num_prop #TopologicalVectorbundleOverCompactHausdorffSpaceIsDirectSummandOfTrivialBundle}
###### Proposition
**(vector bundles over a [[compact Hausdorff space]] are [[direct sum of vector bundles|direct summands]] of a [[trivial vector bundle]])**

Let

1. $X$ be a [[compact Hausdorff space]];

1. $E \to X$ a [[topological vector bundle]] (def. \ref{TopologicalVectorBundle}).

Then there exists another topological vector bundle $\tilde E \to X$ such that the [[direct sum of vector bundles]] of the
two is [[isomorphism|isomorphic]] to a [[trivial vector bundle]] $X \times k^n$:

$$
  E \oplus \tilde E
   \;\simeq\;
  X \times k^n
  \,.
$$

=--

+-- {: .proof}
###### Proof

Let $\{U_i \subset X\}_{i \in I}$ be an [[open cover]] of $X$ over which $E \to X$ has a [[local trivialization]]

$$
  \left\{
    \phi_i
      \;\colon\;
     U_i \times k^n
       \overset{\simeq}{\longrightarrow}
     E\vert_{U_i}
  \right\}_{i \in I}
  \,.
$$

By compactness of $X$, there is a [[finite cover|finite sub-cover]], hence a [[finite set]] $J \subset I$ such tat

$$
  \{U_i \subset X\}_{i \in J \subset I}
$$

is still an open cover over which $E$ trivializes.

Since [[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]] there exists a [[partition of unity]]

$$
  \left\{
    f_i \;\colon\; X \to [0,1]
  \right\}_{i \in J}
$$

with [[support]] $supp(f_i) \subset U_i$. Hence the functions

$$
  \array{
    E\vert_{U_i}
     &\overset{\phantom{AAAA}}{\longrightarrow}&
    U_i \times k^n
    \\
    v
       &\overset{\phantom{AAA}}{\mapsto}&
    f_i(x) \cdot \phi_i^{-1}(v)
  }
$$

extend by 0 to vector bundle homomorphism of the form

$$
  f_i \cdot \phi^{-1}_i
  \;\colon\;
  E
    \longrightarrow
  X \times k^n
  \,.
$$

The finite pointwise [[direct sum]] of these yields a vector bundle homomorphism
of the form

$$
  \underset{i \in J}{\oplus} f_i \cdot \phi_i
  \;\colon\;
  E
    \longrightarrow
  X \times \left(
    \underset{i \in J}{\oplus}
       k^n
  \right)
  \simeq
  X \times k^{n \dot {\vert J\vert}}
  \,.
$$

Observe that, as opposed to the single $f_i \cdot \phi^{-1}_i$, this is a fiber-wise injective, because at each point at least one of the $f_i$ is non-vanishing. Hence this is an injection of $E$ into a trivial vector bundle.

With this the statement follows by prop. \ref{TopologicalSubBundlesOverParacompactHausdorffSpacesAreDirectSummands}.

=--

+-- {: .num_remark}
###### Remark

Prop. \ref{TopologicalVectorbundleOverCompactHausdorffSpaceIsDirectSummandOfTrivialBundle} is key in the analysis of [[topological K-theory]] groups on [[compact Hausdorff spaces]]. See [there](topological+K-theory#DirectSumHasInverseUpToTrivialBundle) for more.

=--

### Concordance
 {#ConcordanceOfTopolgicslVectorBundles}

We discuss that every [[concordance]] of topological vector bundles over a [[paracompact topological space]] makes the restriction of the vector bundle over the endpoints of the interval isomorphic (prop. \ref{ConcondanceOfTopologicalVectorBundles} below). In particular this implies tht the [[pullbacks of vector bundles]] along two [[homotopy|homotopic]] [[continuous functions]] are [[isomorphism|isomorphic]] (corollary \ref{PullbackOfvectorBundlesAlongHomotopicMapsAreIsomorphic} below).

This result is apparently due to Steenrod, see Theorem 11.4 in [Steenrod](#Steenrod).
The proof below follows [Hatcher, theorem 1.6](#Hatcher).

For $X$ a [[topological space]] write $X \times I$ for the [[product topological space]] with the [[closed interval]] $[0,1]$ equipped with its [[Euclidean space|Euclidean]] [[metric topology]].

Write

$$
  X \overset{p_X}{\longleftarrow} X \times [0,1] \overset{p_{[0,1]}}{\longrightarrow} [0,1]
$$

for the two continuous [[projections]] out of the product space.

+-- {: .num_lemma #TrivilizationOfVectorBundleOverProductSpaceWithInterval}
###### Lemma

For $X$ a [[topological space]], then a vector bundle $E \to X \times [0,1]$ is trivializable (example \ref{TrivialTopologicalVectorBundle})
if its restrictions to $X \times [0,1/2]$ and to $X \times [1/2,1]$ are trivializable.

=--

+-- {: .num_lemma #CoverForProductSpaceWithIntrval}
###### Lemma

For $X$ a [[topological space]], then for every topological vector bundle $E \to X \times I$ there exists an [[open cover]] $\{U_i \subset X\}_{i \in I}$ of $X$ such that the vector bundle trivializes over $U_i \times [0,1] \subset X \times [0,1]$, for each $i \in I$.

=--

+-- {: .proof}
###### Proof

By [[local trivialization|local trvializability]] of the vector bundle, there exists an open cover $\{V_j \subset X \times I\}_{j \in J}$ over which the bundle trivializes. For each point $x \in X$ this induces a cover of $\{x\} \times [0,1]$. This is a [[compact topological space]] (for instance by the [[Heine-Borel theorem]]) and hence there exists a [[finite set|finite]] [[subset]] $J_x \subset I$ such that $\{V_i \subset X \times I\}_{i \in J_x}$ still covers $\{x\} \times [0,1]$.

By finiteness of $J_x$, the [[intersection]]

$$
  U_x \coloneqq \underset{i \in J_x}{\cap} p_X(V_i)
$$

is an open neighbourhood of $x$ in $X$. Moreover

$$
  \{ p_{[0,1]}(V_i) \subset I \}_{i \in J_x}
$$

is an open cover of $[0,1]$ such that the given vector bundle trivializes over each element of $\{U_x \times p_{[0,1]}(V_i)\}_{i \in J_x}$.

By the nature of the Euclidean [[metric topology]] each [[open subset]] of $[0,1]$ is a union of intervals. So we may pass to a [[refinement]] of this cover of $[0,1]$ such that each element is a single interval. Again by compactness of $[0,1]$, this refinement has a finite subcover

$$
  \{W_{x,k} \subset [0,1]\}_{k \in K_x}
$$

each element of which is an [[interval]]. Since this is a finite cover, we may find numbers $\{0 = t_0 \lt t_1 \lt t_2 \lt \cdots \lt t_{n_x} = 1\}$ such that

$$
  \{ [t_k, t_{k+1}] \subset [0,1] \}_{0 \leq k \lt n_x}
$$

is a cover of $[0,1]$, and such that the given vector bundle still trivializes over $V_x \times [t_k, t_{k+1}]$ for all $0 \leq k \lt n_x$.

By lemma \ref{TrivilizationOfVectorBundleOverProductSpaceWithInterval} this implies that the vector bundle in fact trivializes over $U_x \times [0,1]$.

Applying this procedure for all points $x \in X$ yields a cover

$$
  \{ U_x \subset X  \}_{x \in X}
$$

with the required property.

=--

+-- {: .num_prop #ConcondanceOfTopologicalVectorBundles}
###### Proposition
**([[concordance]] of [[topological vector bundles]])**

Let $X$ be a [[paracompact Hausdorff space]]. If $E \to X \times [0,1]$ is a [[topological vector bundle]] over the [[product space]] of $X$ with the [[closed interval]] (hence a _[[concordance]]_ of topological vector bundles on $X$), then the two endpoint-restrictions

$$
  E|_{X \times \{0\}}
  \phantom{AA}
  \text{and}
  \phantom{AA}
  E|_{X \times \{1\}}
$$

are [[isomorphism|isomorphic]] vector bundles over $X$.

=--

+-- {: .proof}
###### Proof

By lemma \ref{CoverForProductSpaceWithIntrval} there exists an open cover $\{U_i \subset X\}_{i \in I}$ of $X$ such that the vector bundle $E$ trivializes over $U_i \times [0,1]$ for each $i \in I$. By [this lemma](paracompact+topological+space#CountableCoverOfUnionsofOpenSubsetsInsideGivenCover) there exists a [[countable cover]]

$$
  \{V_n \subset X\}_{n \in \mathbb{N}}
$$

such that each element is a [[disjoint union]] of open subsets that each are contained in one of the $U_i$. This means that the vector bundle $E$ still trivializes over $V_n \times [0,1]$, for each $n \in \mathbb{N}$.

Moreover, since [[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]], there
exists a [[partition of unity]] $\left\{f_n \colon X \to [0,1] \right\}_{n \in \mathbb{N}}$ subordinate to this countable cover.

For $n \in \mathbb{N}$ define

$$
  \psi_n \coloneqq \underoverset{k = 0}{n}{\sum} f_n
$$

(so $\psi_0 = 0$ and by local finiteness there is for each $x \in X$ an $n_x$ such that $\psi_{n \gt n_x} = 1$.)

Now write

$$
  X_n \coloneqq graph( \psi_n ) \subset X \times [0,1]
$$

for the [[graph]] of the function $\psi_n$ equipped with its [[subspace topology]], and write

$$
  E_n
   \coloneqq
  \psi_n^\ast E
$$

for the restriction of $E$ to that subspace

$$
  \array{
    E_n &\longrightarrow& E
    \\
    \downarrow && \downarrow
    \\
    X_n = graph(\psi_n) &\hookrightarrow& X
  }
$$

Observe that the projection functions

$$
  \array{
   p_{n+1,n} \colon &  X_{n+1} &\overset{}{\longrightarrow}& X_n
    \\
    & (x,\psi_{n+1}(x)) &\overset{\phantom{AA}}{\mapsto}& (x, \psi_n(x)) = (x,  \psi_{n+1}(x) - f_{n+1}(x))
  }
$$

are [[continuous functions]]: By the nature of the [[product topology]] and the [[subspace topology]] it is sufficient to check for $U \subset X$
and $V \subset \mathbb{R}$ open subsets, that every point $(x,c)$ in the preimage
$p_n^{-1}( U \times V ) \subset X \times [0,1]$ is contained in an open subset of the form $U_x \times V_x \subset X \times [0,1]$
such that every point of $X_{n+1}$ that is also in $U_x \times V_x$ is still mapped to $U \times V$. Such an open subset is
$\left( U \cap \psi_n^{-1}(V) \right) \times [0,1]$.

Also observe that the composites

$$
  E_n \longrightarrow X_n \overset{p_{n,0}}{\longrightarrow} X_0 = 0
$$

make each $E_n$ a vector bundle over $X$: To see local trivializability over $X$ choose a local
trivialization of $E$ over some open cover $\{U_i \subset X\}_{i \in I}$
and observe that then $E_n$ is trivial over the [[fiber product]]$X_n \times_X U_n$ and hence over $U_n$.

Now by the pullback definition of the $E_n$, the [[pasting law]] says that for each $n \in \mathbb{N}$ we have a
pullback square of vector bundles of the form

$$
  \array{
    E_{n+1}
      && \overset{h_n}{\longrightarrow} &&
    E_n
    \\
    \downarrow && (pb) && \downarrow
    \\
    X_{n+1} && \longrightarrow && X_n
    \\
    & \searrow && \swarrow
    \\
    && X
  }
  \,.
$$

By the nature of pullbacks, the top horizontal function $h_n$ in this diagram is on each fiber a linear isomorphism.
Therefore prop. \ref{FiberwiseIsoisIsomorphismOfVectorBundles} implies that each $h_n$ is in
fact an isomorphism of vector bundles over $X$

By local finiteness, each point $x \in X$ has a neighbourhood $U_x$ such that only a finite number $n_x$ of these $h_n$ are non-trivial, and so it makes sense to consider the infinite composition

$$
  h \coloneqq h_1 \circ h_2 \circ h_3 \circ \cdots
$$

understood to be on each $U_x$ the finite composite

$$
  h(x) \coloneqq h_1 \circ \cdots \circ h_{n_x}
  \,.
$$

Since all the $h_k$ are vector bundle isomorphisms, so are all their composites. Thus $h$ is an isomorphism of the required form

$$
  h
  \;\colon\;
  E|_{X \times \{0\}}
   \overset{\simeq}{\longrightarrow}
  E|_{X \times \{1\}}
  \,.
$$

=--

+-- {: .num_cor #PullbackOfvectorBundlesAlongHomotopicMapsAreIsomorphic}
###### Corollary

Let $X$ be a [[paracompact Hausdorff space]], let $E \to Y$ be a topological [[vector bundle]], let $f,g \colon X \to Y$ be two [[continuous functions]], and let $\eta \colon f \to g$ be a [[left homotopy]] between them. Then there is an [[isomorphism]] of vector bundles over $X$ between the [[pullback of vector bundles]] of $E$ along $f$ and along $g$, respectively:

$$
  f^\ast E \simeq g^\ast E
  \,.
$$

=--

+-- {: .proof}
###### Proof

By definition, the [[left homotopy]] $\eta$ is a [[continuous function]] of the form

$$
  \eta
  \;\colon\;
  X \times [0,1] \longrightarrow Y
  \,.
$$

For $t \in [0,1]$ write $i_t$ for the continuous function

$$
  \array{
     X &\overset{\phantom{AA}i_t\phantom{AA}}{\longrightarrow}& X \times [0,1]
     \\
     x &\overset{\phantom{AAAA}}{\mapsto}& (x,t)
  }
  \,.
$$

By the [[pasting law]] for pullbacks we have that

$$
  f^\ast E = (\eta \circ i_0)^\ast E
  \simeq i_0^\ast (\eta^\ast E) \simeq (\eta^\ast E)|_{X \times \{0\}}
$$

and

$$
  g^\ast E = (\eta \circ i_1)^\ast E
  \simeq i_1^\ast (\eta^\ast E) \simeq (\eta^\ast E)|_{X \times \{1\}}
$$

With this the statement follows by prop. \ref{ConcondanceOfTopologicalVectorBundles}.

=--

+-- {: .num_example #HomotopyInvarianceOfIsomorphismClassesOfVectorBundles}
###### Example
**([[homotopy invariance]] of isomorphism classes of vector bundles)**

Let $X$ and $Y$ be [[paracompact Hausdorff spaces]] and let

$$
  f \;\colon\; X \longrightarrow Y
$$

be a [[continuous function]] which is a [[homotopy equivalence]]. Then
pullback along $f$ constitutes a [[bijection]] on sets of isomorphism classes
of topological vector bundles:

$$
  f^\ast
   \;\colon\;
  Vect(Y)/_\sim
    \overset{\simeq}{\longrightarrow}
  Vect(X)/_\sim
  \,.
$$

=--

+-- {: .proof}
###### Proof

By definition of homotopy equivalence, there is a continuous function
$g \colon Y \longrightarrow X$ and [[left homotopies]]

$$
  g \circ f \Rightarrow id
  \phantom{AAAA}
  f \circ g \Rightarrow id
  \,.
$$

Hence corollary \ref{PullbackOfvectorBundlesAlongHomotopicMapsAreIsomorphic} implies that

$$
  f^\ast \circ g^\ast = (g \circ f)^\ast = id
  \phantom{AAAAA}
  g^\ast \circ f^\ast = (f \circ g)^\ast = id
  \,.
$$

This mean that $g^\ast$ is the [[inverse function]] to $f^\ast$, and hence both are bijections.

=--

+-- {: .num_example #TopologicalVectorBundleOverContractibleSpaceIsTrivializable}
###### Example
**(topological vector bundle on contractible topological space is trivializable)**

If $X$ is a [[contractible topological space]], then every topological vector bundle over $X$
is isomorphic to a [[trivial vector bundle]].

=--

+-- {: .proof}
###### Proof

That $X$ is contractible means by definition that there is a [[left homotopy]] of the form

$$
  \array{
    X &\longrightarrow& \ast
    \\
    \mathllap{i_0}\downarrow & & \downarrow
    \\
    X \times [0,1]
      &\overset{\eta}{\longrightarrow}&
    X
    \\
    \mathllap{i_1}\uparrow & \nearrow_{\mathrlap{id}}
    \\
    X &
  }
  \,.
$$

By cor \ref{PullbackOfvectorBundlesAlongHomotopicMapsAreIsomorphic} it follows that for
$E \to X$ any topological vector bundle that there is an isomorphism between
$id^\ast E = E$ and the result of first restricting the bundle to the point, and then
forming the [[pullback bundle]] along $X \to \ast$. But the latter operation precisely produces
the [[trivial vector bundles]] over $X$.

=--

### Over closed subspaces
 {#OverClosedSubspaces}

We discuss the behavour of vector bundles with respect to [[closed subspaces]] $A \subset X$
of [[compact Hausdorff spaces]].

+-- {: .num_lemma #IsomorphismOfVectorBundlesOnClosedSubsetOfCompactHausdorffSpaceExtendsToOpenNeighbourhoods}
###### Lemma
**([[isomorphism]] of vector bundles on [[closed subset]] of [[compact Hausdorff spaces]] [[extension|extends]] to [[open neighbourhood]])**

Let $k \in \{\mathbb{R}, \mathbb{C}\}$, let $X$ be a [[compact Hausdorff space]] and let $A \subset X$ a [[closed subset|closed]] [[subspace]].
Let $E_i \overset{p_i}{\to} X$ be two topological vector bundles over $X$, $i \in \{1,2\}$.

If there exists an [[isomorphism]]

$$
  E_1\vert_A \overset{\simeq}{\longrightarrow} E_2\vert_A
$$

of the restricted vector bundles over $A$, then there also exists an [[open subset]] $U \subset X$
with $A \subset U$ such that there is also an isomorphism

$$
  E_1\vert_U \overset{\simeq}{\longrightarrow} E_2\vert_U
$$

of the vector bundles restricted to $U$.

=--

+-- {: .proof}
###### Proof

A bundle isomorphism $E_1\vert_A \simeq E_2\vert_A$ is equivalently a trivializing section
(example \ref{FiberwiseLinearlyIndependentSectionsTrivialize}) of the
[[tensor product of vector bundles]] $(E_1\vert_A)^\ast \otimes_A E_2\vert_A$ of $E_2\vert_A$ with the [[dual vector bundle]] $(E_2\vert_A)^\ast$.
(by [this prop.](tensor+product+of+vector+bundles#FinitrRankBundleHomomorphismIsSectionOfTensorProductWithDual)).

Let $\{V_i \subset X\}_{i \in I}$ be an [[open cover]] of $X$ over which this tensor product bundle trivializes with
trivializations

$$
  \left\{
    V_i \times \mathbb{R}^{(n^2)}
     \underoverset{\simeq}{\phi_i}{\longrightarrow}
    (E_1^\ast \otimes_X E_2)\vert_{U_i}
  \right\}
  \,.
$$

Since [[compact Hausdorff spaces are normal]], the [[shrinking lemma]] applies and gives a refinement of this
by a cover $\{U_i \subset X\}_{i \in I}$ by _closed_ subsets $U_i \subset X$.

Then a trivializing section $\sigma \in \Gamma_A\left( (E_1\vert_A)^\ast \otimes_A E_2 \vert_A \right)$ as above is on each $U_i \cap A$
a [[continuous function]]

$$
 \sigma_i
 \;\colon\;
 U_i \cap A \longrightarrow GL(n,k)
$$

to the [[general linear group]] $GL(n,k) \subset Mat_{n \times n}(k)$, such that

$$
  \sigma\vert_{U_i \cap A} = \phi_i \circ \sigma_i
  \,.
$$

Regarded as a function
to the $n \times n$ [[matrices]], this is a set of $n^2$ [[continuous function]] $((\sigma_i)_{a b})$

Now since $U_i \subset X$ is closed by construction, and $A \subset X$ is closed by assumption,
also the intersections $U_i \cap X$ are closed.
Since [[compact Hausdorff spaces are normal]] the [[Tietze extension theorem]] therefore applies to these component functions and yields
[[extensions]] of each $\sigma_i$ to a [[continuous function]] of the form

$$
  \hat \sigma_i \;\colon\; U_i \longrightarrow Mat_{n \times n}(k)
  \,.
$$

Moreover, since compact Hausdorff spaces are evidently [[paracompact Hausdorff spaces]], and since
[[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]], it follows that we find a
[[partition of unity]] $\{f_i \colon U_i \to \mathbb{R} \}_{i \in I}$.

Consider then the functions $f_i \cdot \hat \sigma_i$ given by pointwise multiplication and regarded, via extension by
zero, as continuous functions on all of $X$

$$
  f_i \cdot \hat \sigma_i \;\colon\; X \longrightarrow \mathbb{R}
  \,.
$$

Summing these up yields a single section $\hat \sigma$ of $E_1^\ast \otimes_X E_2$

$$
  \hat \sigma
    \coloneqq
  \sum_{i \in I}
  \phi_i(f_i \cdot \hat \sigma_i)
  \in
  \Gamma_X(E_1^\ast \otimes_X E_2)
  \,,
$$

which by construction is an [[extension]] of the original section, in that

$$
  \hat \sigma\vert_A = \sigma
  \,.
$$

This is because for each $a \in A \subset X$ we have, using the above definitions,

$$
 \begin{aligned}
   \left(\underset{i \in I}{\sum} \phi_i(f_i \cdot \hat \sigma_i)\right)(a)
   & =
   \underset{i \in I}{\sum} (\phi_i (\hat \sigma_i(a)))
   \\
   & = \underset{i \in I}{\sum} \phi_i( f_i(a) \sigma_i(a) )
   \\
   & = \underset{i \in I}{\sum} f_i(a) \cdot (\phi_i \circ \sigma_i)(a)
   \\
   & = \underset{i \in I}{\sum} f_i(a) \cdot \sigma(a)
   \\
   & = \left( \underset{i \in I}{\sum} f_i(a)\right) \cdot \sigma(a)
   \\
   & = \sigma(a)
 \end{aligned}
$$

Here the last step uses the nature of the partition of unity.

Now while $\hat \sigma$ is an extension of the section $\sigma$ to $X$, it will in general
not be a trivializing section on $X$.

But since the [[general linear group]] $GL(n,k) = det^{-1}(k \setminus \{0\}) \subset Mat_{n \times n}(k)$
is an [[open subset]] of the [[Euclidean space]] $Mat_{n \times n}(k) \simeq k^{(n^2)}$, it follows that
each point $x \in A$ has an open neighbourhood $U_x \subset X$ such that $\hat \sigma\vert_{U_x}$
is still a trivializing section, namely choosing $i_x \in I$ such that $x \in U_{i_x}$ set

$$
  U_x \coloneqq (\hat \sigma_{i_x})^{-1}( GL(n,k) )
  \,.
$$

The union of these

$$
  U \coloneqq \underset{x \in A}{\cup} U_x
$$

is hence an open subset containing $A$ such that $(E_1^\ast \otimes_X E_2)\vert_U$ has a trivializing section,
extending $\sigma$, hence such that there is an isomorphism $E_1\vert_U \simeq E_2 \vert_U$ extending the
original isomorphism on $A$.

=--

As a consequence:

+-- {: .num_prop #VectorBundleOnClosedSubsetOfCompactHausdorffSpaceIsPullbackOfBundeOnQuotientSpace}
###### Proposition
**([[vector bundle]] [[trivial vector bundle|trivial]] over [[closed subspace]] of [[compact Hausdorff space]] is [[pullback bundle|pullback]] of bundle on [[quotient space]])**

Let $X$ be a [[compact Hausdorff space]] and let $A \subset X$ be a [[closed subspace]].

If a topological vector bundle $E \overset{p}{\to} X$ is such that its restriction $E\vert_A$
is [[trivializable vector bundle|trivializable]], then $E$ is [[isomorphism|isomorphic]] to
the [[pullback bundle]] $q^\ast E'$ of a topological vector bundle $E' \to X/A$
over the [[quotient space]].

=--

+-- {: .proof}
###### Proof

Let

$$
  A \times k^n
    \underoverset{\simeq}{\phi}{\longrightarrow}
  E\vert_A
$$

be an isomorphism of vector bundles over $A$, which exists by assumption.
Consider then on the total space $E\vert_A$ the [[equivalence relation]] given by

$$
  \phi^{-1}(x,v) \sim \phi^{-1}(x',v)
$$

for all $x,x' \in A$ and $v \in k^n$. Let

$$
  E' \coloneqq E/\sim
$$

be the corresponding [[quotient topological space]]. Observe that for $x \in X$
we have $E'_x = E_x$ while for $x \in A$ we have a canonical identification $E'_{x/A} \simeq k^n$,
and over these points quotient coprojection is identified with $\phi^{-1}$:

$$
  \array{
    E
      &\overset{}{\longrightarrow}&
    E'
    \\
    (x,v) &\mapsto& \left\{ \array{ (x,v) &\vert& x \in X \setminus A  \\ \phi^{-1}_x(v) &\vert& x\in A }  \right.
  }
  \,.
$$

Since the composite continuous function

$$
  E \overset{p}{\longrightarrow} X \overset{q}{\longrightarrow} X/A
$$

respects the equivalence relation (in that it sends any two equivalent points to the same image point)
the [[universal property]] of the quotient space yields
a continuous function

$$
  p' \;\colon\; E' \to X/A
$$

such that the following [[commuting diagram|diagram commutes]]

$$
  \array{
    E &\longrightarrow& E'
    \\
    {}^{\mathllap{p}}\downarrow && \downarrow^{\mathrlap{p'}}
    \\
    X &\overset{q}{\longrightarrow}& X/A
  }
  \,.
$$

We claim that this is a [[pullback]] diagram in [[Top]]:

By the above description of the top horizontal function, it is a pullback diagram
of underlying sets. Hence we need to see that the topology on $E$
has a [[base for a topology|base]] given by the pre-images
of the open subsets in $X$ and in $E'$. Now by definition of the quotient space
topology on $E'$, its open subsets are those of $E$ that either do not
contain a point $(x,v)$ with $x \in A$ or if they do, then they also
contain all the points of the form $(x', \phi_{x'}^{-1}(\phi_x(v)))$ for
$x' \in A$. Moreover, if $(x,v)$ is in the open subset for $x \in A$,
then also $(x,v')$ for all $v'$ in some open ball in $k^n$ containing $v$.
Hence intersecing these pre-images with pre-images of open subsets of $X$ under $p$
yields a basis for the topology.

Hence it only remains to see that $E' \overset{p'}{\longrightarrow} X/A$ is a vector bundle. The
fiberwise linearity is clear, we need to show that it is locally trivializable.

To that end, let $\{U_i \subset X\}_{i \in I}$ be an open cover over which $E \overset{p}{\to} X$
has a local trivialization. Since $A \subset X$ is assumed to be closed, it follows that

$$
  \left\{ U_i \setminus A \subset X \setminus A\right\}_{i \in I}
$$

is an open cover of the complement of $A$ in $X$. By the nature of the [[quotient space topology]], this
induces an open cover of $X\setminus A$. If we adjoin the quotient $U/A$  of an open neighbourhood $U \subset X$ of $A$ in $X$,
then

$$
  \{ U_i \setminus A \subset X/A \} \sqcup \{ U/A \subset X/A \}
$$

is an open cover of $X/A$. Moreover, by the construction of $E' \overset{p'}{\to} X/A$ it is clear that
this bundle has a local trivialization over $U_i$, since $E \overset{p}{\to} X$ does,
and similarly $E'$ trivializes over $U/A$ if $E$ trivializes over $U$. But such a $U$ does indeed exist by
lemma \ref{IsomorphismOfVectorBundlesOnClosedSubsetOfCompactHausdorffSpaceExtendsToOpenNeighbourhoods}.

=--

+-- {: .num_remark}
###### Remark

Prop \ref{VectorBundleOnClosedSubsetOfCompactHausdorffSpaceIsPullbackOfBundeOnQuotientSpace} is the reason why [[reduced K-theory|reduced]] [[topological K-theory]]
satisfies the [[long exact sequences in cohomology]] that make it a [[generalized (Eilenberg-Steenrod) cohomology theory]].
See

=--

+-- {: .num_prop #VectorBundlesOverQuotientByContractibleSubspaceAreEquivalentToVectorBundlesOnTotalSpace}
###### Proposition

Let $X$ be a [[compact Hausdorff space]] and $A \subset X$ a [[closed subset|closed]] [[subspace]]
and write $X/A$ for the corresponding [[quotient topological space]] ([this example](quotient+space#QuotientBySubspace))
with quotient coprojection denoted
$q \colon X \longrightarrow X/A$.

If $A$ is a [[contractible topological space]] then the [[pullback bundle]] construction

$$
  q^\ast \;\colon\; Vect(X/A)_{/\sim} \longrightarrow Vect(X)_{/\sim}
$$

is an [[isomorphism]].

=--

+-- {: .proof}
###### Proof

By example \ref{TopologicalVectorBundleOverContractibleSpaceIsTrivializable} every vector bundle
$E \overset{p}{\to} X$ is trivializable over the contractible subspace $A$.
Therefore prop. \ref{VectorBundleOnClosedSubsetOfCompactHausdorffSpaceIsPullbackOfBundeOnQuotientSpace}
implies that it is in the image of the pullback bundle map $q^\ast$. This says that $q^\ast$
is surjective. Finally, it is clear that it is injective. Therefore it is bijective.

=--

+-- {: .num_example }
###### Example

Let $(X,x)$ be a [[pointed topological space|pointed]] [[compact topological space]].

For $[0,1] \subset \mathbb{R}$ the [[closed interval]] with its [[Euclidean space|Euclidean]] [[metric topology]].

There is

1. the ordinary [[cylinder]], being the [[product space]] $X \times I$

1. the [[reduced cylinder]] $X \wedge I_+ = (X \times I)/( \{x\} \times I )$
   which is the [[smash product]] with the interval that has a base point freely adjoined

and

1. the ordinary [[suspension]] $S X \coloneqq (X \times I)/( X \times \{0,1\} )$;

1. the [[reduced suspension]] $\Sigma X \coloneqq (S X)/( \{x\} \times I )$.

In both cases the reduced space is obtained from the unreduced space by quotienting out the
contractible closed subspace $I \simeq \{x\} \times I$ and hence topological vector bundles
do not see the difference between the reduced and the unreduced spaces, by prop. \ref{VectorBundlesOverQuotientByContractibleSubspaceAreEquivalentToVectorBundlesOnTotalSpace}.

=--

## Related concepts

* [[algebraic vector bundle]]

* [[differentiable vector bundle]]

* [[topological K-theory]]

## References

The original reference for many results about bundles, including the theorem that [[concordance]] implies [[isomorphism]], is

* {#Steenrod} [[Norman Steenrod]], _The Topology of Fibre Bundles_, Princeton University Press (1951, 1957, 1960) [[jstor:j.ctt1bpm9t5](https://www.jstor.org/stable/j.ctt1bpm9t5)]

Further textbook accounts:

* {#MilnorStasheff74} [[John Milnor]], [[Jim Stasheff]], _Characteristic classes_, Princeton Univ. Press (1974) [[ISBN:9780691081229](https://press.princeton.edu/books/paperback/9780691081229/characteristic-classes-am-76-volume-76), [doi:10.1515/9781400881826](https://doi.org/10.1515/9781400881826), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/milnstas.pdf)]

* Glenys Luke, Alexander S. Mishchenko, _Vector bundles and their applications_, Math. and its Appl. __447__, Kluwer 1998. viii+254 pp. [MR99m:55019](http://www.ams.org/mathscinet-getitem?mr=99m:55019)

* [[Dale Husemoeller]], [[Michael Joachim]], [[Branislav Jurco]], [[Martin Schottenloher]], _[[Basic Bundle Theory and K-Cohomology Invariants]]_, Lecture Notes in Physics, Springer 2008 ([pdf](http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726corr1.pdf))

Lecture notes with an eye towards [[topological K-theory]]:

* {#Wirthmuller12} [[Klaus Wirthmüller]], _Vector bundles and K-theory_, 2012 ([[wirthmueller-vector-bundles-and-k-theory.pdf:file]])

* {#Hatcher} [[Allen Hatcher]], chapter 1 of _Vector bundles and K-Theory_, (partly finished book) [web](http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html)

[[!redirects topological vector bundles]]


topological vector space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

A **topological vector space**, or **TVS** for short, is a [[vector space]] $X$ over a [[topological field]] (usually a [[local field]], more often than not the field of [[real numbers]] or the field of [[complex numbers]] with the usual topology) $k$ (called the [[ground field]]) equipped with a [[topological space|topology]] for which the addition and scalar multiplication maps
$$+: X \times X \to X, \qquad \cdot: k \times X \to X$$
are [[continuous function|continuous]]. The topological vector spaces over a given field form a category [[TopVect]].

Much as a [[topological group]] is a [[group object]] in [[Top]], so a TVS is the same as a vector space [[internalization|internal to]] $Top$ ... *provided* that we use the _two-sorted_ notion of vector space $(k, X)$ so that the first sort is interpreted as the topological ground field. (More generally, there is a notion of topological module which is the internalization in $Top$ of the two-sorted notion of module involving a ring sort and a module over that ring; this notion is purely equational and thus interpretable in terms of finite products. Then a TVS is a topological module whose underlying ring is a field.) N.B.: if we internalize the one-sorted notion vector space, in which the ground field is encoded in the [[Lawvere theory]] as the set of unary operations, the continuity of the unary operations gives only a continuous map ${|k|} \times X \to X$ where we use the [[discrete topology]] on $k$ -- which is not what we want.

Like any topological [[abelian group]], a TVS $X$ carries a [[uniform space]] structure generated by a basis of [[entourages]] (aka vicinities) that correspond to neighborhoods $U$ of $0$:
$$\{(u, v) \in X \times X: u - v \in U\}$$

Thus many uniform notions (uniform continuity, completeness, etc.) carry over to the TVS context. Also from the uniformity (although it is also easy to prove directly), it follows that a TVS is completely regular, and also Hausdorff if and only if it is $T_0$ (see [[separation axiom]]). Most authors insist on the $T_0$ condition to rule out degenerate cases, but that prevents the category of TVSes from being [[topological concrete category|topological]] over [[Vect]].
If the TVS $V$ is not Hausdorff, then the subset $V_0$ defined as the intersection of all neighborhoods of zero is a vector subspace of $V$ and the [[quotient space|quotient vector space]] $V/V_0$ is Hausdorff, hence Tihonov (= completely regular Hausdorff).

The condition that scalar multiplication is continuous puts significant constraints on the topology of $X$. For example, if we assume $k$ has a base of compact absorbing neighborhoods of $0$ and $V$ is Hausdorff, then for any non-zero $v \in X$ the function
$$- \cdot v: k \to V$$
maps $k$ [[homeomorphism|homeomorphically]] onto its image. It can then be shown that $X$ cannot (for instance) be [[compact space|compact]] (unless it is the zero space and so has no non-zero $v$); a classical theorem along these lines is that $V$ (over a [[local field]]) can be locally compact Hausdorff if and only if $V$ is finite-dimensional. (In the non-Hausdorff case, the theorems are that $X$ is compact if and only if its topology is indiscrete and that $X$ is locally compact if and only if it is a finitary [[direct sum]] of indiscrete spaces.) On the other hand, a nice property of even infinite-dimensional TVSes is that they are [[path-connected space|path-connected]], or at least so in the classical cases where the ground field is $\mathbb{R}$ or $\mathbb{C}$.

More classical material should be added, particularly on [[locally convex spaces]].

### TVSes from a Hilbert space viewpoint

The theory of TVS can be understood as the quest to find the essence of many fundamental theorems of [[functional analysis]] of [[Hilbert space]]s (or [[Banach space]]s), namely to find the minimal set of assumptions that are needed for Hilbert space theorems to remain true.
Examples of these are:

* the [Hahn--Banach Theorem](http://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem)

* the [Open Mapping](http://en.wikipedia.org/wiki/Open_mapping_theorem_%28functional_analysis%29) and [[closed graph theorem]]s

A central rôle in the whole theory plays **duality**, that is the study of [[locally convex spaces]] and their duals. A prominent example is the definition of certain concepts by duality in the theory of [[Schwartz distribution]]s.

## Important subclasses
 {#ImportantSubclasses}

Topological vector spaces come in many flavours. The following chart
provides a first overview (chart originally created and published by
[[Greg Kuperberg]] on MathOverflow
[here](http://mathoverflow.net/questions/8443/barrelled-bornological-ultrabornological-semi-reflexive-how-are-these-use),
current version generated using Graphviz from [[lctvs dot source]]):

[[!include TVS relationships]]

- [[locally convex spaces]]: where the Hahn-Banach theorem works (assuming sufficient axioms)

- [[bornological topological vector space]]s: where bounded means continuous

- [[barrelled topological vector space]]s

* [[sequence space]]

* [[stereotype space]]

## Related concepts

* [[topology of uniform convergence]]

* [[topological vector bundle]]

## References

* [[Nicolas Bourbaki]], *Topological Vector Spaces*, Chapters 1–5, Masson (1981), Springer (2003) [[doi:10.1007/978-3-642-61715-7](https://doi.org/10.1007/978-3-642-61715-7)]

* {#Treves67} [[François Trèves]], _Topological Vector Spaces, Distributions and Kernels_ (Academic Press, New York, 1967)

The following review gives lots of important examples

* Paul Garrett, _Catalogue of Useful Topological Vectorspaces_, 2011 ([pdf](http://www-users.math.umn.edu/~garrett/m/fun/catalogue_tvs.pdf))

See also

* Wikipedia, _[Topological vector space](http://en.wikipedia.org/wiki/Topological_vector_space)_

category: analysis

[[!redirects topological vector space]]
[[!redirects topological vector spaces]]
[[!redirects TVS]]
[[!redirects TVSs]]
[[!redirects TVSes]]


topological Yang-Mills theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Topological Yang-Mills theory is a [[gauge theory]] [[topological quantum field theory]] .

## Definition

For $X$ a 4-[[dimensional]] [[smooth manifold]], $\mathfrak{g}$ a [[Lie algebra]] with [[Lie group]] $G$ and $\langle-,-\rangle \in W(\mathfrak{g})$ a binary [[invariant polynomial]] on $\mathfrak{g}$, **topological Yang-Mills theory** is the [[quantum field theory]] defined by the [[action functional]]

$$
  S : G Bund_\nabla(X) \to \mathbb{R}
$$

on the [[groupoid]] of $G$-[[principal bundle]]s with [[connection on a bundle]] that sends a connecton $\nabla$ to the integral of the [[curvature]] 4-form $\langle F_\nabla \wedge F_\nabla \rangle$ of the corresponding [[Chern-Simons circle 3-bundle]]:

$$
  S : \nabla \mapsto \int_X \langle F_A \wedge F_A\rangle
  \,.
$$

## Relation to other models

The ordinary kinetic term of [[Yang-Mills theory]] differs from this by the fact that the [[Hodge star operator]] appears $\langle F_\nabla \wedge \star F_\nabla \rangle$. In full Yang-Mills theory both terms appear.

The topological Yang-Mills action also appears in the [[generalized Chern-Simons theory]] given by a [[Chern-Simons element]] in a [[Lie 2-algebra]], where it is coupled to [[BF-theory]]. See [[Chern-Simons element]] for details.

## Related concepts

* [[theta angle]]

* [[topologically twisted D=4 super Yang-Mills theory]]

## References

The term originates with

* L. Baulieu, [[Isadore Singer]], _Topological Yang-Mills symmetry_, Nuclear Physics B - Proceedings Supplements Volume 5, Issue 2, December 1988, Pages 12-19 (doi:10.1016/0920-5632(88)90366-0)

following

* {#Witten88} [[Edward Witten]], _Topological quantum field theory_, Comm. Math. Phys. Volume 117, Number 3 (1988), 353-386. ([euclid:1104161738](http://projecteuclid.org/euclid.cmp/1104161738))

Review emphasizing the relation to [[Chern-Simons theory]] is

* P van Baal, _An introduction to topological Yang-Mills theory_, Acta Phys.Polon. B21 (1990) 73 ([spire:280417](http://inspirehep.net/record/280417), [pdf](http://www.actaphys.uj.edu.pl/fulltext?series=Reg&vol=21&page=73))

The relation to [[Chern-Simons theory]] on the boundary in an ambient [[string theory|string theoretic]] context is indicated in section 2 (starting around p. 21) of

* {#Witten} [[Edward Witten]], _Fivebranes and Knots_ ([arXiv:1101.3216](http://arxiv.org/abs/1101.3216))

On the [[Gribov ambiguity]] in topological Yang-Mills theory:

* D. Dudal, C. P. Felix, O. C. Junqueira, D. S. Montes, A. D. Pereira, G. Sadovski, R. F. Sobreiro, A. A. Tomaz, _Gribov problem in topological Yang-Mills theories_ ([arXiv:1907.05460](https://arxiv.org/abs/1907.05460))

See also

* O. C. Junqueira, A. D. Pereira, G. Sadovski, R. F. Sobreiro, A. A. Tomaz, _Absence of radiative corrections in 4-dimensional topological Yang-Mills theories_, Phys. Rev. D 98, 021701 (2018) ([arXiv:1805.01850](https://arxiv.org/abs/1805.01850))

[[!redirects topological Yang-Mills theories]]


topologically complete space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
=--
=--

# Topologically complete spaces
* table of contents
{: toc}

## Idea

A [[topological space]] is topologically complete if and only if it is [[complete space|completely]] [[metrizable space|metrizable]].  However, the term 'topologically complete' may be applied to other [[spaces]] as well, and in this case it means that the [[underlying topological space]] is completely metrizable.

One can vary this by considering some other kind of underlying space (besides the underlying topological one) and other kinds of completability (besides complete metrizability).

## Definitions

### General

Let $C$, $M$, $S$, and $T$ be [[categories]], and suppose that we have [[functors]] $U\colon C \to M$, $V\colon M \to S$, and $W\colon S \to T$.  We will think of the [[objects]] of these categories as [[spaces]], and we\'ll think of these functors and their [[composites]] as [[forgetful functors]].  Also, we\'ll think of $C$ as a [[subcategory]] of $M$, and we\'ll think of the objects of $C$ as the _complete_ spaces in $M$.

The basic example to keep in mind is the case where $T$ is the category [[Top]] of [[topological spaces]] and [[continuous functions]], $M$ is the category [[Met]] of [[metric spaces]] and [[short functions]], and $C$ is the [[full subcategory]] of $Met$ consisting of the [[complete metric spaces]].  In the motivating example, $S$ is also $Met$, but this can remain flexible.  In any case, $U, V, W$ should all be standard forgetful functors.

Even in other situations, it should be that once you specify $M, S, T$, then $C, U, V, W$ should all be obvious.  Typically, $C, M, S, T$ will be [[concrete categories]], $U, V, W$ will be [[faithful functor|faithful]] [[isofibrations]], and $U$ will additionally be [[full functor|full]], but this doesn\'t really affect anything.

An object $X$ of $S$ is __$M$-izable__ if there is an object $Y$ of $M$ and an [[isomorphism]] in $S$ between $X$ and $V(Y)$.  $X$ is __completely $M$-izable__ if there is an object $Z$ of $C$ and an isomorphism in $S$ between $X$ and $V(U(Z)$.  $X$ is __$T$-ly $M$-izable__ if there is an object $Y$ of $M$ and an isomorphism in $T$ between $W(X)$ and $W(V(Y))$.  Finally, $X$ is __$T$-ly completely $M$-izable__ if there is an object $Z$ of $C$ and an isomorphism in $T$ between $W(X)$ and $W(V(U(Z)))$.  (If all functors are isofibrations, then we may without loss of generality require these $T$-isomorphisms to be [[equalities]], which is typically what is done in topology books.)

The actual terminology used will be based on terms for the objects of $M$ and $T$ rather than terms for $M$ and $T$ themselves; for example, in the standard example where $M$ is $Met$ and $T$ is $Top$, then we say that $X$ is __topologically completely metrizable__ (to mean that $X$ is $Top$-ly completely $Met$-izable).  $C, U, V, W$ are suppressed in the terminology because they are supposed to be obvious; $S$ is suppressed because you are supposed to know what it is when you first mention $X$.

### Special cases

A space $X$ (an object of $S$) is $M$-izable if and only if $X$ is $S$-ly $M$-izable; $X$ is completely $M$-izable iff $X$ is $S$-ly completely $M$-izable.  $X$ is [[complete space|complete]] iff $X$ is $S$-ly completely $S$-izable; $X$ is always $S$-ly $S$-izable.  Finally, $X$ is $T$-ly completely $M$-izable iff $X$ is $T$-ly $C$-izable.

A space is __topologically complete__ if it is topologically completely metrizable; this is the standard example above.  At least one reference ([Schechter 1997](#Schechter1997)) uses this term for a space that is topologically completely *pseudometrizable* (that is, $M$ is the category of [[pseudometric spaces]]).  According to Wikipedia, at least one reference ([Kelley 1955](#Kelley1955)) uses this term for a space that is topologically completely *uniformizable* (that is, $M$ is the category of [[uniform spaces]]).

A space is __Dieudonné-complete__ if it is topologically completely uniformizable.  At least one reference ([Arkhangel′skii 1977](#Ark1977)) uses this term for a space that is topologically completely *Hausdorff*-uniformizable (that is, $M$ is the category of of [[Hausdorff space|Hausdorff]] uniform spaces).

## Categories of complete spaces

If we consider the entire category of complete spaces in some sense, then $S$ must be specified explicitly; that is, we speak of the category of $T$-ly $M$-izable $S$-spaces or the category of $T$-ly completely $M$-izable $S$-spaces.  However, if $S$ is not specified, then we take it to be $T$ by default, so the $T$-ly [completely] $M$-izable spaces are the same as the [completely] $M$-izable $T$-spaces.  We consider these categories to be [[full subcategories]] of $S$.

Up to [[equivalence of categories|equivalence]] of categories, the category of $M$-izable $T$-spaces (aka the $T$-ly $M$-izable spaces, that is the $T$-ly $M$-izable $T$-spaces) is the category whose [[objects]] are taken from $M$ and whose [[morphisms]] are taken from $T$; similarly, the category of completely $M$-izable $T$-spaces has objects taken from $C$ and morphisms taken from $T$.

In particular, following our standard example, the category of topologically complete spaces (that is the category of completely metrizable toplogical spaces) may be taken to have complete metric spaces as objects and continuous functions as morphisms.

## Related concepts

* [[metrisable space]]

* [[cotopology]]

* [[uniform space]]

## References

*  A.V. Arkhangel′skii (1977).  _Complete space_.  Matematicheskaya entsiklopediya.  [Updated English version](https://www.encyclopediaofmath.org/index.php/Complete_space).
   {#Ark1977}

*  [[John Kelley]] (1955).  _General Topology_, 1975.
   {#Kelley1955}

*  Eric Schechter (1997).  _[[Handbook of Analysis and its Foundations]]_, Section 20.12.
   {#Schechter1997}

[[!redirects topologically complete space]]
[[!redirects topologically complete spaces]]

[[!redirects Dieudonne complete space]]
[[!redirects Dieudonne complete spaces]]
[[!redirects Dieudonné complete space]]
[[!redirects Dieudonné complete spaces]]
[[!redirects Dieudonne-complete space]]
[[!redirects Dieudonne-complete spaces]]
[[!redirects Dieudonné-complete space]]
[[!redirects Dieudonné-complete spaces]]

[[!redirects complete topological space]]
[[!redirects complete topological spaces]]


topologically cyclic group

A [[topological group]] is called **topologically cyclic** or **monothetic** if there exists a [[dense subspace|dense]] [[cyclic group|cyclic]] [[subgroup]].

###Examples

* Any finite cyclic group
* The group of [[p-adic integers]]
* Any torus $(S^1)^n$

[[!redirects topologically cyclic groups]]
[[!redirects monothetic group]]