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 $\R^n.$

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 $\R^2,$ but there is no local homeomorphism $S^2 \to \R^2.$

Formal definition

A function $f : X \to Y$ between two topological spaces is called a 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 $\R \to S^1$ defined by $t \mapsto e^$ (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 $X$ 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 $f : U \to \R^n$ is a continuous injective map from an open subset $U$ of $\R^n,$ then $f(U)$ is open in $\R^n$ and $f : U \to f(U)$ is a homeomorphism.
Consequently, a continuous map $f : U \to \R^n$ from an open subset $U \subseteq \R^n$ 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 $f : U \to \Complex$ (where $U$ is an open subset of the complex plane $\Complex$) is a local homeomorphism precisely when the derivative $f^(z)$ 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 $f : U \to \R^n$ (where $U$ is an open subset of $\R^n$) 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 $f : \R \to \R$ 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.The assumptions that $f$ is continuous and open imply that the set $O = O_f$ is equal to the union of all open subsets $U$ of $X$ such that the restriction $f\big\vert_U : U \to Y$ is an injective map.
If every fiber of $f$ is a discrete subspace of $X$ then this open set $O$ is necessarily a 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 footnoteConsider the continuous open surjection $f : \R \times \R \to \R$ defined by $f(x, y) = x.$ The set $O = O_f$ for this map is the empty set; that is, there does not exist any non-empty open subset $U$ of $\R \times \R$ for which the restriction $f\big\vert_U : U \to \R$ is an injective map. 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 : \R \to inverse function theorem