Local homeomorphism

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 ${\displaystyle \mathbb{R}\to S^1}$ defined by ${\displaystyle t\mapsto e^{it}}$ (so that geometrically, this map wraps the real line around the circle) is a local homeomorphism but not a homeomorphism. The map ${\displaystyle f:S^1\to S^1}$ defined by ${\displaystyle f(z)=z^n,}$ which wraps the circle around itself ${\displaystyle n}$ times (that is, has winding number ${\displaystyle n}$), is a local homeomorphism for all non-zero ${\displaystyle n,}$ but it is a homeomorphism only when it is bijective (that is, only when ${\displaystyle n=1}$ or ${\displaystyle n=-1}$).

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 ${\displaystyle p:X\to Y}$ is a proper local homeomorphism between two Hausdorff spaces and if ${\displaystyle Y}$ is also locally compact, then ${\displaystyle p}$ is a covering map.

Local homeomorphisms and composition of functions

The composition of two local homeomorphisms is a local homeomorphism; explicitly, 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. The restriction of a local homeomorphism to any open subset of the domain will again be a local homomorphism; explicitly, if ${\displaystyle f:X\to Y}$ is a local homeomorphism then its restriction ${\displaystyle f{|}_U:U\to Y}$ to any ${\displaystyle U}$ open subset of ${\displaystyle X}$ is also a local homeomorphism.

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

Inclusion maps

If ${\displaystyle U\subseteq X}$ is any subspace (where as usual, ${\displaystyle U}$ is equipped with the subspace topology induced by ${\displaystyle X}$) then the inclusion map ${\displaystyle i:U\to X}$ is always a topological embedding. But it is a local homeomorphism if and only if ${\displaystyle U}$ is open in ${\displaystyle X.}$ The subset ${\displaystyle U}$ being open in ${\displaystyle X}$ is essential for the inclusion map to be a local homeomorphism because the inclusion map of a non-open subset of ${\displaystyle X}$ never yields a local homeomorphism (since it will not be an open map).

The restriction ${\displaystyle f{|}_U:U\to Y}$ of a function ${\displaystyle f:X\to Y}$ to a subset ${\displaystyle U\subseteq X}$ is equal to its composition with the inclusion map ${\displaystyle i:U\to X;}$ explicitly, ${\displaystyle f{|}_U=f\circ i.}$ Since the composition of two local homeomorphisms is a local homeomorphism, if ${\displaystyle f:X\to Y}$ and ${\displaystyle i:U\to X}$ are local homomorphisms then so is ${\displaystyle f{|}_U=f\circ i.}$ Thus restrictions of local homeomorphisms to open subsets are local homeomorphisms.

Invariance of domain

Invariance of domain guarantees that if ${\displaystyle f:U\to {\mathbb{R}}^n}$ is a continuous injective map from an open subset ${\displaystyle U}$ of ${\displaystyle {\mathbb{R}}^n,}$ then ${\displaystyle f(U)}$ is open in ${\displaystyle {\mathbb{R}}^n}$ and ${\displaystyle f:U\to f(U)}$ is a homeomorphism. Consequently, a continuous map ${\displaystyle f:U\to {\mathbb{R}}^n}$ from an open subset ${\displaystyle U\subseteq {\mathbb{R}}^n}$ will be a local homeomorphism if and only if it is a locally injective map (meaning that every point in ${\displaystyle U}$ has a neighborhood ${\displaystyle N}$ such that the restriction of ${\displaystyle f}$ to ${\displaystyle N}$ is injective).

Local homeomorphisms in analysis

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 ${\displaystyle f^{\mathrm{\prime }}(z)}$ 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 ${\displaystyle n\ge 2.}$ 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.

Local homeomorphisms and fibers

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. If every fiber of ${\displaystyle f}$ is a discrete subspace of ${\displaystyle X}$ (which is a necessary condition for ${\displaystyle f:X\to Y}$ to be a local homeomorphism) then ${\displaystyle f}$ is a ${\displaystyle Y}$-valued local homeomorphism on a dense open subset of ${\displaystyle X.}$ To clarify this statement's conclusion, let ${\displaystyle O=O_f}$ be the (unique) largest open subset of ${\displaystyle X}$ such that ${\displaystyle f{|}_O:O\to Y}$ is a local homeomorphism. If every fiber of ${\displaystyle f}$ is a discrete subspace of ${\displaystyle X}$ then this open set ${\displaystyle O}$ is necessarily a dense subset of ${\displaystyle X.}$ In particular, if ${\displaystyle X\ne \varnothing }$ then ${\displaystyle O\ne \varnothing ;}$ a conclusion that may be false without the assumption that ${\displaystyle f}$'s fibers are discrete (see this footnote for an example). One corollary is that every continuous open surjection ${\displaystyle f}$ between completely metrizable second-countable spaces that has discrete fibers is "almost everywhere" a local homeomorphism (in the topological sense that ${\displaystyle O_f}$ is a dense open subset of its domain). For example, the map ${\displaystyle f:\mathbb{R}\to [0,\mathrm{\infty })}$ defined by the polynomial ${\displaystyle f(x)=x^2}$ is a continuous open surjection with discrete fibers so this result guarantees that the maximal open subset ${\displaystyle O_f}$ is dense in ${\displaystyle \mathbb{R};}$ with additional effort (using the inverse function theorem for instance), it can be shown that ${\displaystyle O_f=\mathbb{R}\setminus \{0\},}$ which confirms that this set is indeed dense in ${\displaystyle \mathbb{R}.}$ This example also shows that it is possible for ${\displaystyle O_f}$ to be a proper dense subset of ${\displaystyle f}$'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 ${\displaystyle f:X\to Y}$ where ${\displaystyle Y}$ is a Hausdorff space but ${\displaystyle X}$ is not. Consider for instance the quotient space ${\displaystyle X=\left(\mathbb{R}\bigsqcup \mathbb{R}\right)/\sim ,}$ 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 ${\displaystyle f^{-1}(\{y\})}$ has two elements if ${\displaystyle y\ge 0}$ and one element if ${\displaystyle y<0.}$ Similarly, it is possible to construct a local homeomorphisms ${\displaystyle f:X\to Y}$ where ${\displaystyle X}$ is Hausdorff and ${\displaystyle Y}$ is not: pick the natural map from ${\displaystyle X=\mathbb{R}\bigsqcup \mathbb{R}}$ to ${\displaystyle Y=\left(\mathbb{R}\bigsqcup \mathbb{R}\right)/\sim }$ with the same equivalence relation ${\displaystyle \sim }$ 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 ${\displaystyle f:X\to Y}$ is a local homeomorphism depends on its codomain. The image ${\displaystyle f(X)}$ of a local homeomorphism ${\displaystyle f:X\to Y}$ is necessarily an open subset of its codomain ${\displaystyle Y}$ and ${\displaystyle f:X\to f(X)}$ will also be a local homeomorphism (that is, ${\displaystyle f}$ will continue to be a local homeomorphism when it is considered as the surjective map ${\displaystyle f:X\to f(X)}$ onto its image, where ${\displaystyle f(X)}$ has the subspace topology inherited from ${\displaystyle Y}$). However, in general it is possible for ${\displaystyle f:X\to f(X)}$ to be a local homeomorphism but ${\displaystyle f:X\to Y}$ to not be a local homeomorphism (as is the case with the map ${\displaystyle f:\mathbb{R}\to {\mathbb{R}}^2}$ defined by ${\displaystyle f(x)=(x,0),}$ for example). A map ${\displaystyle f:X\to Y}$ is a local homomorphism if and only if ${\displaystyle f:X\to f(X)}$ is a local homeomorphism and ${\displaystyle f(X)}$ is an open subset of ${\displaystyle Y.}$

Every fiber of a local homeomorphism ${\displaystyle f:X\to Y}$ is a discrete subspace of its domain ${\displaystyle X.}$

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.

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.