# Initial topology

In general topology and related areas of mathematics, the initial topology (or induced topology or strong topology or limit topology or projective topology) on a set $X,$ with respect to a family of functions on $X,$ is the coarsest topology on $X$ that makes those functions continuous.
The subspace topology and product topology constructions are both special cases of initial topologies. Indeed, the initial topology construction can be viewed as a generalization of these.
The dual notion is the final topology, which for a given family of functions mapping to a set $Y$ is the finest topology on $Y$ that makes those functions continuous.

## Definition

Given a set $X$ and an indexed family $\left(Y_{i}\right)_{i\in I}$ of topological spaces with functions $f_{i}:X\to Y_{i},$ the initial topology $\tau$ on $X$ is the coarsest topology on $X$ such that each $f_{i}:(X,\tau )\to Y_{i}$ is continuous.
Definition in terms of open sets
If $\left(\tau _{i}\right)_{i\in I}$ is a family of topologies $X$ indexed by $I\neq \varnothing ,$ then the least upper bound topology of these topologies is the coarsest topology on $X$ that is finer than each $\tau _{i}.$ This topology always exists and it is equal to the topology generated by ${\textstyle \bigcup \limits _{i\in I}\tau _{i}}.$
If for every $i\in I,$ $\sigma _{i}$ denotes the topology on $Y_{i},$ then $f_{i}^{-1}\left(\sigma _{i}\right)=\left\{f_{i}^{-1}(V):V\in \sigma _{i}\right\}$ is a topology on $X$, and the initial topology of the $Y_{i}$ by the mappings $f_{i}$ is the least upper bound topology of the $I$-indexed family of topologies $f_{i}^{-1}\left(\sigma _{i}\right)$ (for $i\in I$). Explicitly, the initial topology is the collection of open sets generated by all sets of the form $f_{i}^{-1}(U),$ where $U$ is an open set in $Y_{i}$ for some $i\in I,$ under finite intersections and arbitrary unions.
Sets of the form $f_{i}^{-1}(V)$ are often called cylinder sets. If $I$ contains exactly one element, then all the open sets of the initial topology $(X,\tau )$ are cylinder sets.

## Examples

Several topological constructions can be regarded as special cases of the initial topology.
* The subspace topology is the initial topology on the subspace with respect to the inclusion map.
* The product topology is the initial topology with respect to the family of projection maps.
* The inverse limit of any inverse system of spaces and continuous maps is the set-theoretic inverse limit together with the initial topology determined by the canonical morphisms.
* The weak topology on a locally convex space is the initial topology with respect to the continuous linear forms of its dual space.
* Given a family of topologies $\left\{\tau _{i}\right\}$ on a fixed set $X$ the initial topology on $X$ with respect to the functions $\operatorname {id} _{i}:X\to \left(X,\tau _{i}\right)$ is the supremum (or join) of the topologies $\left\{\tau _{i}\right\}$ in the lattice of topologies on $X.$ That is, the initial topology $\tau$ is the topology generated by the union of the topologies $\left\{\tau _{i}\right\}.$
* A topological space is completely regular if and only if it has the initial topology with respect to its family of (bounded) real-valued continuous functions.
* Every topological space $X$ has the initial topology with respect to the family of continuous functions from $X$ to the Sierpiński space.

## Properties
### Characteristic property

The initial topology on $X$ can be characterized by the following characteristic property: A function $g$ from some space $Z$ to $X$ is continuous if and only if $f_{i}\circ g$ is continuous for each $i\in I.$
Note that, despite looking quite similar, this is not a universal property. A categorical description is given below.
A filter ${\mathcal {B}}$ on $X$ converges to a point $x\in X$ if and only if the prefilter $f_{i}({\mathcal {B}})$ converges to $f_{i}(x)$ for every $i\in I.$

### Evaluation

By the universal property of the product topology, we know that any family of continuous maps $f_{i}:X\to Y_{i}$ determines a unique continuous map ${\begin{alignedat}{4}f:\;&&X&&\;\to \;&\prod _{i}Y_{i}\\[0.3ex]&&x&&\;\mapsto \;&\left(f_{i}(x)\right)_{i\in I}\\\end{alignedat}}$
This map is known as the evaluation map.
A family of maps $\{f_{i}:X\to Y_{i}\}$ is said to separate points in $X$ if for all $x\neq y$ in $X$ there exists some $i$ such that $f_{i}(x)\neq f_{i}(y).$ The family $\{f_{i}\}$ separates points if and only if the associated evaluation map $f$ is injective.
The evaluation map $f$ will be a topological embedding if and only if $X$ has the initial topology determined by the maps $\{f_{i}\}$ and this family of maps separates points in $X.$

### Hausdorffness

If $X$ has the initial topology induced by $\left\{f_{i}:X\to Y_{i}\right\}$ and if every $Y_{i}$ is Hausdorff, then $X$ is a Hausdorff space if and only if these maps separate points on $X.$

### Transitivity of the initial topology

If $X$ has the initial topology induced by the $I$-indexed family of mappings $\left\{f_{i}:X\to Y_{i}\right\}$ and if for every $i\in I,$ the topology on $Y_{i}$ is the initial topology induced by some $J_{i}$-indexed family of mappings $\left\{g_{j}:Y_{i}\to Z_{j}\right\}$ (as $j$ ranges over $J_{i}$), then the initial topology on $X$ induced by $\left\{f_{i}:X\to Y_{i}\right\}$ is equal to the initial topology induced by the ${\textstyle \bigcup \limits _{i\in I}J_{i}}$-indexed family of mappings $\left\{g_{j}\circ f_{i}:X\to Z_{j}\right\}$ as $i$ ranges over $I$ and $j$ ranges over $J_{i}.$ Several important corollaries of this fact are now given.
In particular, if $S\subseteq X$ then the subspace topology that $S$ inherits from $X$ is equal to the initial topology induced by the inclusion map $S\to X$ (defined by $s\mapsto s$). Consequently, if $X$ has the initial topology induced by $\left\{f_{i}:X\to Y_{i}\right\}$ then the subspace topology that $S$ inherits from $X$ is equal to the initial topology induced on $S$ by the restrictions $\left\{\left.f_{i}\right|_{S}:S\to Y_{i}\right\}$ of the $f_{i}$ to $S.$
The product topology on $\prod _{i}Y_{i}$ is equal to the initial topology induced by the canonical projections $\operatorname {pr} _{i}:\left(x_{k}\right)_{k\in I}\mapsto x_{i}$ as $i$ ranges over $I.$ Consequently, the initial topology on $X$ induced by $\left\{f_{i}:X\to Y_{i}\right\}$ is equal to the inverse image of the product topology on $\prod _{i}Y_{i}$ by the evaluation map f:X→∏$_i$Y$_i$.{\textstyle f:X\to \prod _{i}Y_{i}\, .} Furthermore, if the maps $\left\{f_{i}\right\}_{i\in I}$ separate points on $X$ then the evaluation map is a homeomorphism onto the subspace $f(X)$ of the product space $\prod _{i}Y_{i}.$

### Separating points from closed sets

If a space $X$ comes equipped with a topology, it is often useful to know whether or not the topology on $X$ is the initial topology induced by some family of maps on $X.$ This section gives a sufficient (but not necessary) condition.
A family of maps $\left\{f_{i}:X\to Y_{i}\right\}$ separates points from closed sets in $X$ if for all closed sets $A$ in $X$ and all $x\not \in A,$ there exists some $i$ such that $f_{i}(x)\notin \operatorname {cl} (f_{i}(A))$ where $\operatorname {cl}$ denotes the closure operator.
    Theorem. A family of continuous maps $\left\{f_{i}:X\to Y_{i}\right\}$ separates points from closed sets if and only if the cylinder sets $f_{i}^{-1}(V),$ for $V$ open in $Y_{i},$ form a base for the topology on $X.$
It follows that whenever $\left\{f_{i}\right\}$ separates points from closed sets, the space $X$ has the initial topology induced by the maps $\left\{f_{i}\right\}.$ The converse fails, since generally the cylinder sets will only form a subbase (and not a base) for the initial topology.
If the space $X$ is a T$_0$ space, then any collection of maps $\left\{f_{i}\right\}$ that separates points from closed sets in $X$ must also separate points. In this case, the evaluation map will be an embedding.

### Initial uniform structure

If $\left({\mathcal {U}}_{i}\right)_{i\in I}$ is a family of uniform structures on $X$ indexed by $I\neq \varnothing ,$ then the least upper bound uniform structure of $\left({\mathcal {U}}_{i}\right)_{i\in I}$ is the coarsest uniform structure on $X$ that is finer than each ${\mathcal {U}}_{i}.$ This uniform always exists and it is equal to the filter on $X\times X$ generated by the filter subbase ${\textstyle \bigcup \limits _{i\in I}{\mathcal {U}}_{i}}.$ If $\tau _{i}$ is the topology on $X$ induced by the uniform structure ${\mathcal {U}}_{i}$ then the topology on $X$ associated with least upper bound uniform structure is equal to the least upper bound topology of $\left(\tau _{i}\right)_{i\in I}.$
Now suppose that $\left\{f_{i}:X\to Y_{i}\right\}$ is a family of maps and for every $i\in I,$ let ${\mathcal {U}}_{i}$ be a uniform structure on $Y_{i}.$ Then the initial uniform structure of the $Y_{i}$ by the mappings $f_{i}$ is the unique coarsest uniform structure ${\mathcal {U}}$ on $X$ making all $f_{i}:\left(X,{\mathcal {U}}\right)\to \left(Y_{i},{\mathcal {U}}_{i}\right)$ uniformly continuous. It is equal to the least upper bound uniform structure of the $I$-indexed family of uniform structures $f_{i}^{-1}\left({\mathcal {U}}_{i}\right)$ (for $i\in I$). The topology on $X$ induced by ${\mathcal {U}}$ is the coarsest topology on $X$ such that every $f_{i}:X\to Y_{i}$ is continuous. The initial uniform structure ${\mathcal {U}}$ is also equal to the coarsest uniform structure such that the identity mappings $\operatorname {id} :\left(X,{\mathcal {U}}\right)\to \left(X,f_{i}^{-1}\left({\mathcal {U}}_{i}\right)\right)$ are uniformly continuous.
Hausdorffness: The topology on $X$ induced by the initial uniform structure ${\mathcal {U}}$ is Hausdorff if and only if for whenever $x,y\in X$ are distinct ($x\neq y$) then there exists some $i\in I$ and some entourage $V_{i}\in {\mathcal {U}}_{i}$ of $Y_{i}$ such that $\left(f_{i}(x),f_{i}(y)\right)\not \in V_{i}.$ Furthermore, if for every index $i\in I,$ the topology on $Y_{i}$ induced by ${\mathcal {U}}_{i}$ is Hausdorff then the topology on $X$ induced by the initial uniform structure ${\mathcal {U}}$ is Hausdorff if and only if the maps $\left\{f_{i}:X\to Y_{i}\right\}$ separate points on $X$ (or equivalently, if and only if the evaluation map f:X→∏$_i$Y$_i${\textstyle f:X\to \prod _{i}Y_{i}} is injective)
Uniform continuity: If ${\mathcal {U}}$ is the initial uniform structure induced by the mappings $\left\{f_{i}:X\to Y_{i}\right\},$ then a function $g$ from some uniform space $Z$ into $(X,{\mathcal {U}})$ is uniformly continuous if and only if $f_{i}\circ g:Z\to Y_{i}$ is uniformly continuous for each $i\in I.$
Cauchy filter: A filter ${\mathcal {B}}$ on $X$ is a Cauchy filter on $(X,{\mathcal {U}})$ if and only if $f_{i}\left({\mathcal {B}}\right)$ is a Cauchy prefilter on $Y_{i}$ for every $i\in I.$
Transitivity of the initial uniform structure: If the word "topology" is replaced with "uniform structure" in the statement of "transitivity of the initial topology" given above, then the resulting statement will also be true.

## Categorical description

In the language of category theory, the initial topology construction can be described as follows. Let $Y$ be the functor from a discrete category $J$ to the category of topological spaces $\mathrm {Top}$ which maps $j\mapsto Y_{j}$. Let $U$ be the usual forgetful functor from $\mathrm {Top}$ to $\mathrm {Set}$. The maps $f_{j}:X\to Y_{j}$ can then be thought of as a cone from $X$ to $UY.$ That is, $(X,f)$ is an object of $\mathrm {Cone} (UY):=(\Delta \downarrow {UY})$—the category of cones to $UY.$ More precisely, this cone $(X,f)$ defines a $U$-structured cosink in $\mathrm {Set} .$
The forgetful functor $U:\mathrm {Top} \to \mathrm {Set}$ induces a functor ${\bar {U}}:\mathrm {Cone} (Y)\to \mathrm {Cone} (UY)$. The characteristic property of the initial topology is equivalent to the statement that there exists a universal morphism from ${\bar {U}}$ to $(X,f);$ that is, a terminal object in the category $\left({\bar {U}}\downarrow (X,f)\right).$ Explicitly, this consists of an object $I(X,f)$ in $\mathrm {Cone} (Y)$ together with a morphism $\varepsilon :{\bar {U}}I(X,f)\to (X,f)$ such that for any object $(Z,g)$ in $\mathrm {Cone} (Y)$ and morphism $\varphi :{\bar {U}}(Z,g)\to (X,f)$ there exists a unique morphism $\zeta :(Z,g)\to I(X,f)$ such that the following diagram commutes:
The assignment $(X,f)\mapsto I(X,f)$ placing the initial topology on $X$ extends to a functor $I:\mathrm {Cone} (UY)\to \mathrm {Cone} (Y)$ which is right adjoint to the forgetful functor ${\bar {U}}.$ In fact, $I$ is a right-inverse to ${\bar {U}}$; since ${\bar {U}}I$ is the identity functor on $\mathrm {Cone} (UY).$