# Local homeomorphism

In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If $f:X\to Y$ is a local homeomorphism, $X$ is said to be an étale space over $Y.$ Local homeomorphisms are used in the study of sheaves. Typical examples of local homeomorphisms are covering maps.
A topological space $X$ is locally homeomorphic to $Y$ if every point of $X$ has a neighborhood that is homeomorphic to an open subset of $Y.$ For example, a manifold of dimension $n$ is locally homeomorphic to $\mathbb {R} ^{n}.$
If there is a local homeomorphism from $X$ to $Y,$ then $X$ is locally homeomorphic to $Y,$ but the converse is not always true. For example, the two dimensional sphere, being a manifold, is locally homeomorphic to the plane $\mathbb {R} ^{2},$ but there is no local homeomorphism $S^{2}\to \mathbb {R} ^{2}.$

## Formal definition

A function $f:X\to Y$ between two topological spaces is called a local homeomorphism if every point $x\in X$ has an open neighborhood $U$ whose image $f(U)$ is open in $Y$ and the restriction $f{\big \vert }_{U}:U\to f(U)$ is a homeomorphism (where the respective subspace topologies are used on $U$ and on $f(U)$).

## Examples and sufficient conditions

Local homeomorphisms versus homeomorphisms
Every homeomorphism is a local homeomorphism. But a local homeomorphism is a homeomorphism if and only if it is bijective. A local homeomorphism need not be a homeomorphism. For example, the function $\mathbb {R} \to S^{1}$ defined by $t\mapsto e^{it}$ (so that geometrically, this map wraps the real line around the circle) is a local homeomorphism but not a homeomorphism. The map $f:S^{1}\to S^{1}$ defined by $f(z)=z^{n},$ which wraps the circle around itself $n$ times (that is, has winding number $n$), is a local homeomorphism for all non-zero $n,$ but it is a homeomorphism only when it is bijective (that is, only when $n=1$ or $n=-1$).
Generalizing the previous two examples, every covering map is a local homeomorphism; in particular, the universal cover $p:C\to Y$ of a space $Y$ is a local homeomorphism. In certain situations the converse is true. For example: if $p:X\to Y$ is a proper local homeomorphism between two Hausdorff spaces and if $Y$ is also locally compact, then $p$ is a covering map.
Local homeomorphisms and composition of functions
The composition of two local homeomorphisms is a local homeomorphism; explicitly, if $f:X\to Y$ and $g:Y\to Z$ are local homeomorphisms then the composition $g\circ f:X\to Z$ is also a local homeomorphism. The restriction of a local homeomorphism to any open subset of the domain will again be a local homomorphism; explicitly, if $f:X\to Y$ is a local homeomorphism then its restriction $f{\big \vert }_{U}:U\to Y$ to any $U$ open subset of $X$ is also a local homeomorphism.
If $f:X\to Y$ is continuous while both $g:Y\to Z$ and $g\circ f:X\to Z$ are local homeomorphisms, then $f$ is also a local homeomorphism.
Inclusion maps
If $U\subseteq X$ is any subspace (where as usual, $U$ is equipped with the subspace topology induced by $X$) then the inclusion map $i:U\to X$ is always a topological embedding. But it is a local homeomorphism if and only if $U$ is open in $X.$ The subset $U$ being open in $X$ is essential for the inclusion map to be a local homeomorphism because the inclusion map of a non-open subset of $X$ never yields a local homeomorphism (since it will not be an open map).
The restriction $f{\big \vert }_{U}:U\to Y$ of a function $f:X\to Y$ to a subset $U\subseteq X$ is equal to its composition with the inclusion map $i:U\to X;$ explicitly, $f{\big \vert }_{U}=f\circ i.$ Since the composition of two local homeomorphisms is a local homeomorphism, if $f:X\to Y$ and $i:U\to X$ are local homomorphisms then so is $f{\big \vert }_{U}=f\circ i.$ Thus restrictions of local homeomorphisms to open subsets are local homeomorphisms.
Invariance of domain
Invariance of domain guarantees that if $f:U\to \mathbb {R} ^{n}$ is a continuous injective map from an open subset $U$ of $\mathbb {R} ^{n},$ then $f(U)$ is open in $\mathbb {R} ^{n}$ and $f:U\to f(U)$ is a homeomorphism. Consequently, a continuous map $f:U\to \mathbb {R} ^{n}$ from an open subset $U\subseteq \mathbb {R} ^{n}$ will be a local homeomorphism if and only if it is a locally injective map (meaning that every point in $U$ has a neighborhood $N$ such that the restriction of $f$ to $N$ is injective).
Local homeomorphisms in analysis
It is shown in complex analysis that a complex analytic function $f:U\to \mathbb {C}$ (where $U$ is an open subset of the complex plane $\mathbb {C}$) is a local homeomorphism precisely when the derivative $f^{\prime }(z)$ is non-zero for all $z\in U.$ The function $f(x)=z^{n}$ on an open disk around $0$ is not a local homeomorphism at $0$ when $n\geq 2.$ In that case $0$ is a point of "ramification" (intuitively, $n$ sheets come together there).
Using the inverse function theorem one can show that a continuously differentiable function $f:U\to \mathbb {R} ^{n}$ (where $U$ is an open subset of $\mathbb {R} ^{n}$) is a local homeomorphism if the derivative $D_{x}f$ is an invertible linear map (invertible square matrix) for every $x\in U.$ (The converse is false, as shown by the local homeomorphism $f:\mathbb {R} \to \mathbb {R}$ with $f(x)=x^{3}$). An analogous condition can be formulated for maps between differentiable manifolds.
Local homeomorphisms and fibers
Suppose $f:X\to Y$ is a continuous open surjection between two Hausdorff second-countable spaces where $X$ is a Baire space and $Y$ is a normal space. If every fiber of $f$ is a discrete subspace of $X$ (which is a necessary condition for $f:X\to Y$ to be a local homeomorphism) then $f$ is a $Y$-valued local homeomorphism on a dense open subset of $X.$ To clarify this statement's conclusion, let $O=O_{f}$ be the (unique) largest open subset of $X$ such that $f{\big \vert }_{O}:O\to Y$ is a local homeomorphism. If every fiber of $f$ is a discrete subspace of $X$ then this open set $O$ is necessarily a dense subset of $X.$ In particular, if $X\neq \varnothing$ then $O\neq \varnothing ;$ a conclusion that may be false without the assumption that $f$'s fibers are discrete (see this footnote for an example). One corollary is that every continuous open surjection $f$ between completely metrizable second-countable spaces that has discrete fibers is "almost everywhere" a local homeomorphism (in the topological sense that $O_{f}$ is a dense open subset of its domain). For example, the map $f:\mathbb {R} \to [0,\infty )$ defined by the polynomial $f(x)=x^{2}$ is a continuous open surjection with discrete fibers so this result guarantees that the maximal open subset $O_{f}$ is dense in $\mathbb {R} ;$ with additional effort (using the inverse function theorem for instance), it can be shown that $O_{f}=\mathbb {R} \setminus \{0\},$ which confirms that this set is indeed dense in $\mathbb {R} .$ This example also shows that it is possible for $O_{f}$ to be a proper dense subset of $f$'s domain. Because every fiber of every non-constant polynomial is finite (and thus a discrete, and even compact, subspace), this example generalizes to such polynomials whenever the mapping induced by it is an open map.
Local homeomorphisms and Hausdorffness
There exist local homeomorphisms $f:X\to Y$ where $Y$ is a Hausdorff space but $X$ is not. Consider for instance the quotient space $X=\left(\mathbb {R} \sqcup \mathbb {R} \right)/\sim ,$ where the equivalence relation $\sim$ on the disjoint union of two copies of the reals identifies every negative real of the first copy with the corresponding negative real of the second copy. The two copies of $0$ are not identified and they do not have any disjoint neighborhoods, so $X$ is not Hausdorff. One readily checks that the natural map $f:X\to \mathbb {R}$ is a local homeomorphism. The fiber $f^{-1}(\{y\})$ has two elements if $y\geq 0$ and one element if $y<0.$ Similarly, it is possible to construct a local homeomorphisms $f:X\to Y$ where $X$ is Hausdorff and $Y$ is not: pick the natural map from $X=\mathbb {R} \sqcup \mathbb {R}$ to $Y=\left(\mathbb {R} \sqcup \mathbb {R} \right)/\sim$ with the same equivalence relation $\sim$ as above.

## Properties

A map is a local homeomorphism if and only if it is continuous, open, and locally injective. In particular, every local homeomorphism is a continuous and open map. A bijective local homeomorphism is therefore a homeomorphism.
Whether or not a function $f:X\to Y$ is a local homeomorphism depends on its codomain. The image $f(X)$ of a local homeomorphism $f:X\to Y$ is necessarily an open subset of its codomain $Y$ and $f:X\to f(X)$ will also be a local homeomorphism (that is, $f$ will continue to be a local homeomorphism when it is considered as the surjective map $f:X\to f(X)$ onto its image, where $f(X)$ has the subspace topology inherited from $Y$). However, in general it is possible for $f:X\to f(X)$ to be a local homeomorphism but $f:X\to Y$ to not be a local homeomorphism (as is the case with the map $f:\mathbb {R} \to \mathbb {R} ^{2}$ defined by $f(x)=(x,0),$ for example). A map $f:X\to Y$ is a local homomorphism if and only if $f:X\to f(X)$ is a local homeomorphism and $f(X)$ is an open subset of $Y.$
Every fiber of a local homeomorphism $f:X\to Y$ is a discrete subspace of its domain $X.$
A local homeomorphism $f:X\to Y$ transfers "local" topological properties in both directions:
* $X$ is locally connected if and only if $f(X)$ is;
* $X$ is locally path-connected if and only if $f(X)$ is;
* $X$ is locally compact if and only if $f(X)$ is;
* $X$ is first-countable if and only if $f(X)$ is.
As pointed out above, the Hausdorff property is not local in this sense and need not be preserved by local homeomorphisms.
The local homeomorphisms with codomain $Y$ stand in a natural one-to-one correspondence with the sheaves of sets on $Y;$ this correspondence is in fact an equivalence of categories. Furthermore, every continuous map with codomain $Y$ gives rise to a uniquely defined local homeomorphism with codomain $Y$ in a natural way. All of this is explained in detail in the article on sheaves.

## Generalizations and analogous concepts

The idea of a local homeomorphism can be formulated in geometric settings different from that of topological spaces. For differentiable manifolds, we obtain the local diffeomorphisms; for schemes, we have the formally étale morphisms and the étale morphisms; and for toposes, we get the étale geometric morphisms.