Local homeomorphism

Let ${\displaystyle X}$ and ${\displaystyle Y}$ be topological spaces. A function ${\displaystyle f:X\to Y}$ is a local homeomorphism[1] if for every point ${\displaystyle x\in X}$ there exists an open set ${\displaystyle U}$ containing ${\displaystyle x,}$ such that the image ${\displaystyle f(U)}$ is open in ${\displaystyle Y}$ and the restriction ${\displaystyle f{\big \vert }_{U}:U\to X}$ is a homeomorphism (where the respective subspace topologies are used on ${\displaystyle U}$ and on ${\displaystyle f(U)}$).

By definition, every homeomorphism is also a local homeomorphism.

If ${\displaystyle U}$ is an open subset of ${\displaystyle Y}$ equipped with the subspace topology, then the inclusion map ${\displaystyle i:U\to Y}$ is a local homeomorphism. Openness is essential here: the inclusion map of a non-open subset of ${\displaystyle Y}$ never yields a local homeomorphism.

Let ${\displaystyle f:\mathbb {R} \to S^{1}}$ be the map that wraps the real line around the circle (i.e. ${\displaystyle f(t)=e^{it}}$ for all ${\displaystyle t\in \mathbb {R} .}$ This is a local homeomorphism but not a homeomorphism.

Let ${\displaystyle f:S^{1}\to S^{1}}$ be the map that wraps the circle around itself ${\displaystyle n}$ times (i.e. has winding number ${\displaystyle n}$). This is a local homeomorphism for all non-zero ${\displaystyle n,}$ but a homeomorphism only in the cases where it is bijective, i.e. when ${\displaystyle n=1}$ or ${\displaystyle -1.}$

Generalizing the previous two examples, every covering map is a local homeomorphism; in particular, the universal cover ${\displaystyle p:C\to Y}$ of a space ${\displaystyle Y}$ is a local homeomorphism. In certain situations the converse is true. For example: if ${\displaystyle X}$ is Hausdorff and ${\displaystyle Y}$ is locally compact and Hausdorff and ${\displaystyle p:X\to Y}$ is a proper local homeomorphism, then ${\displaystyle p}$ is a covering map.

There are local homeomorphisms ${\displaystyle f:X\to Y}$ where ${\displaystyle Y}$ is a Hausdorff space and ${\displaystyle X}$ is not. Consider for instance the quotient space ${\displaystyle X=\left(\mathbb {R} \sqcup \mathbb {R} \right)/\sim ,}$ where the equivalence relation ${\displaystyle \sim }$ 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 ${\displaystyle 0}$ are not identified and they do not have any disjoint neighborhoods, so ${\displaystyle X}$ is not Hausdorff. One readily checks that the natural map ${\displaystyle f:X\to \mathbb {R} }$ is a local homeomorphism. The fiber ${\displaystyle f^{-1}\left(\{y\}\right)}$ has two elements if ${\displaystyle y\geq 0}$ and one element if ${\displaystyle y<0.}$

Similarly, we can construct a local homeomorphisms ${\displaystyle f:X\to Y}$ where ${\displaystyle X}$ is Hausdorff and ${\displaystyle Y}$ is not: pick the natural map from ${\displaystyle X=\mathbb {R} \sqcup \mathbb {R} }$ to ${\displaystyle Y=\left(\mathbb {R} \sqcup \mathbb {R} \right)/\sim }$ with the same equivalence relation ${\displaystyle \sim }$ as above.

If ${\displaystyle f:X\to Y}$ is a local homeomorphism and ${\displaystyle U}$ is an open subset of ${\displaystyle X,}$ then the restriction ${\displaystyle f{\big \vert }_{U}:U\to Y}$ is also a local homeomorphism.

If ${\displaystyle f:X\to Y}$ and ${\displaystyle g:Y\to Z}$ are local homeomorphisms, then the composition ${\displaystyle g\circ f:X\to Z}$ is also a local homeomorphism.

If ${\displaystyle f:X\to Y}$ is continuous, ${\displaystyle g:Y\to Z}$ is a local homeomorphism, and ${\displaystyle g\circ f:X\to Z}$ a local homeomorphism, then ${\displaystyle f}$ is also a local homeomorphism.

It is shown in complex analysis that a complex analytic function ${\displaystyle f:U\to \mathbb {C} }$ (where ${\displaystyle U}$ is an open subset of the complex plane ${\displaystyle \mathbb {C} }$) is a local homeomorphism precisely when the derivative ${\displaystyle f^{\prime }(z)}$ is non-zero for all ${\displaystyle z\in U.}$ The function ${\displaystyle f(x)=z^{n}}$ on an open disk around ${\displaystyle 0}$ is not a local homeomorphism at ${\displaystyle 0}$ when ${\displaystyle n\geq 2.}$ In that case ${\displaystyle 0}$ is a point of "ramification" (intuitively, ${\displaystyle n}$ sheets come together there).

Using the inverse function theorem one can show that a continuously differentiable function ${\displaystyle f:U\to \mathbb {R} ^{n}}$ (where ${\displaystyle U}$ is an open subset of ${\displaystyle \mathbb {R} ^{n}}$) is a local homeomorphism if the derivative ${\displaystyle D_{x}f}$ is an invertible linear map (invertible square matrix) for every ${\displaystyle x\in U.}$ (The converse is false, as shown by the local homeomorphism ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ with ${\displaystyle f(x)=x^{3}}$). An analogous condition can be formulated for maps between differentiable manifolds.

Suppose ${\displaystyle f:X\to Y}$ is a continuous open surjection between two Hausdorff second countable spaces where ${\displaystyle X}$ is a Baire space and ${\displaystyle Y}$ is a normal space. Let ${\displaystyle O}$ be the union of all open subsets ${\displaystyle U}$ of ${\displaystyle X}$ such that ${\displaystyle f{\big \vert }_{U}:U\to Y}$ is an injective map, which also makes ${\displaystyle O}$ is the largest open subset of ${\displaystyle X}$ such that ${\displaystyle f{\big \vert }_{O}:O\to Y}$ is a local homeomorphism. If every fiber of ${\displaystyle f}$ is a discrete subspace of ${\displaystyle X}$ then ${\displaystyle O}$ is a dense subset of ${\displaystyle X.}$

Every local homeomorphism is a continuous and open map. A bijective local homeomorphism is therefore a homeomorphism.

Every fiber of a surjective local homeomorphism ${\displaystyle f:X\to Y}$ is a discrete subspace of ${\displaystyle X}$ (and if the empty set with its unique topology is considered to be a discrete space, then this true of all local homeomorphisms, including those that are not surjective).

A local homeomorphism ${\displaystyle f:X\to Y}$ transfers "local" topological properties in both directions:

• ${\displaystyle X}$ is locally connected if and only if ${\displaystyle f(X)}$ is;
• ${\displaystyle X}$ is locally path-connected if and only if ${\displaystyle f(X)}$ is;
• ${\displaystyle X}$ is locally compact if and only if ${\displaystyle f(X)}$ is;
• ${\displaystyle X}$ is first-countable if and only if ${\displaystyle f(X)}$ 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 ${\displaystyle Y}$ stand in a natural one-to-one correspondence with the sheaves of sets on ${\displaystyle Y;}$ this correspondence is in fact an equivalence of categories. Furthermore, every continuous map with codomain ${\displaystyle Y}$ gives rise to a uniquely defined local homeomorphism with codomain ${\displaystyle Y}$ in a natural way. All of this is explained in detail in the article on sheaves.

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.

• Homeomorphism – Isomorphism of topological spaces in mathematics
• Local diffeomorphism

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