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 $$t \mapsto e^{it}$$ (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 $$X$$ 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 $$f^{\prime}(z)$$ 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 $$f : U \to \R^n$$ (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. 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 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 $$f : \R \to [0, \infty)$$ 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 $$\R;$$ with additional effort (using the inverse function theorem for instance), it can be shown that $$O_f = \R \setminus \{0\},$$ which confirms that this set is indeed dense in $$\R.$$ This example also shows that it is possible for $$O_f$$ to be a 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 $$X = \left(\R \sqcup \R\right) / \sim,$$ 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 $$f : X \to \R$$ 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 $$X = \R \sqcup \R$$ to $$Y = \left(\R \sqcup \R\right) / \sim$$ 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 be a local homeomorphism (as is the case with the map $$f : \R \to \R^2$$ 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.