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 → Y {\displaystyle f:X\to Y} is a local homeomorphism, X {\displaystyle X} is said to be an étale space over Y . {\displaystyle Y.} Local homeomorphisms are used in the study of sheaves. Typical examples of local homeomorphisms are covering maps.

A topological space X {\displaystyle X} is locally homeomorphic to Y {\displaystyle Y} if every point of X {\displaystyle X} has a neighborhood that is homeomorphic to an open subset of Y . {\displaystyle Y.} For example, a manifold of dimension n {\displaystyle n} is locally homeomorphic to R n . {\displaystyle \mathbb {R} ^{n}.}

If there is a local homeomorphism from X {\displaystyle X} to Y , {\displaystyle Y,} then X {\displaystyle X} is locally homeomorphic to Y , {\displaystyle 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 , {\displaystyle \mathbb {R} ^{2},} but there is no local homeomorphism S 2 → R 2 . {\displaystyle S^{2}\to \mathbb {R} ^{2}.}

Formal definition

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

Examples and sufficient conditions

Covering maps

Every homeomorphism is a local homeomorphism. The function R → S 1 {\displaystyle \mathbb {R} \to S^{1}} defined by t ↦ e i t {\displaystyle t\mapsto e^{it}} (so that geometrically, this map wraps the real line around the circle in the complex plane) is a local homeomorphism but not a homeomorphism. The map f : S 1 → S 1 {\displaystyle f:S^{1}\to S^{1}} defined by f ( z ) = z n , {\displaystyle f(z)=z^{n},} where n {\displaystyle n} is a fixed integer, wraps the circle around itself n {\displaystyle n} times (that is, has winding number n {\displaystyle n}) and is a local homeomorphism for all non-zero n , {\displaystyle n,} but it is a homeomorphism only when it is bijective (that is, only when n = 1 {\displaystyle n=1} or n = − 1 {\displaystyle n=-1}).

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

Inclusion maps of open subsets

If U ⊆ X {\displaystyle U\subseteq X} is any subspace (where as usual, U {\displaystyle U} is equipped with the subspace topology induced by X {\displaystyle X}) then the inclusion map i : U → X {\displaystyle i:U\to X} is always a topological embedding. It is a local homeomorphism if and only if U {\displaystyle U} is open in X . {\displaystyle X.}

Invariance of domain

Invariance of domain guarantees that if f : U → R n {\displaystyle f:U\to \mathbb {R} ^{n}} is a continuous injective map from an open subset U {\displaystyle U} of R n , {\displaystyle \mathbb {R} ^{n},} then f ( U ) {\displaystyle f(U)} is open in R n {\displaystyle \mathbb {R} ^{n}} and f : U → f ( U ) {\displaystyle f:U\to f(U)} is a homeomorphism. Consequently, a continuous map f : U → R n {\displaystyle f:U\to \mathbb {R} ^{n}} from an open subset U ⊆ R n {\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 U {\displaystyle U} has a neighborhood N {\displaystyle N} such that the restriction of f {\displaystyle f} to N {\displaystyle N} is injective).

Local homeomorphisms in analysis

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

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

Local homeomorphisms and Hausdorffness

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

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

Local homeomorphisms and fibers

Suppose f : X → Y {\displaystyle f:X\to Y} is a continuous open surjection between two Hausdorff second-countable spaces where X {\displaystyle X} is a Baire space and Y {\displaystyle Y} is a normal space. If every fiber of f {\displaystyle f} is a discrete subspace of X {\displaystyle X} (which is a necessary condition for f : X → Y {\displaystyle f:X\to Y} to be a local homeomorphism) then f {\displaystyle f} is a Y {\displaystyle Y}-valued local homeomorphism on a dense open subset of X . {\displaystyle X.} To clarify this statement's conclusion, let O = O f {\displaystyle O=O_{f}} be the (unique) largest open subset of X {\displaystyle X} such that f | O : O → Y {\displaystyle f{\big \vert }_{O}:O\to Y} is a local homeomorphism. If every fiber of f {\displaystyle f} is a discrete subspace of X {\displaystyle X} then this open set O {\displaystyle O} is necessarily a dense subset of X . {\displaystyle X.} In particular, if X ≠ ∅ {\displaystyle X\neq \varnothing } then O ≠ ∅ ; {\displaystyle O\neq \varnothing ;} a conclusion that may be false without the assumption that f {\displaystyle f}'s fibers are discrete (see this footnote for an example). One corollary is that every continuous open surjection f {\displaystyle f} between completely metrizable second-countable spaces that has discrete fibers is "almost everywhere" a local homeomorphism (in the topological sense that O f {\displaystyle O_{f}} is a dense open subset of its domain). For example, the map f : R → [ 0 , ∞ ) {\displaystyle f:\mathbb {R} \to [0,\infty )} defined by the polynomial f ( x ) = x 2 {\displaystyle f(x)=x^{2}} is a continuous open surjection with discrete fibers so this result guarantees that the maximal open subset O f {\displaystyle O_{f}} is dense in R ; {\displaystyle \mathbb {R} ;} with additional effort (using the inverse function theorem for instance), it can be shown that O f = R ∖ { 0 } , {\displaystyle O_{f}=\mathbb {R} \setminus \{0\},} which confirms that this set is indeed dense in R . {\displaystyle \mathbb {R} .} This example also shows that it is possible for O f {\displaystyle O_{f}} to be a proper dense subset of f {\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.

Properties

A map f : X → Y {\displaystyle f:X\to Y} is a local homeomorphism if and only if it is continuous, open, and locally injective (the latter means that every point in X {\displaystyle X} has a neighborhood N {\displaystyle N} such that the restriction of f {\displaystyle f} to N {\displaystyle N} is injective). It follows that the map f {\displaystyle f} is a homeomorphism if and only if it is a bijective local homeomorphism.

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

Whether or not a function is a local homeomorphism depends on its codomain: A map f : X → Y {\displaystyle f:X\to Y} is a local homeomorphism if and only if the surjection f : X → f ( X ) {\displaystyle f:X\to f(X)} is a local homeomorphism (where f ( X ) {\displaystyle f(X)} has the subspace topology inherited from Y {\displaystyle Y}) and f ( X ) {\displaystyle f(X)} is an open subset of Y . {\displaystyle Y.}

Local homeomorphisms and composition of functions

The composition of two local homeomorphisms is a local homeomorphism; explicitly, if f : X → Y {\displaystyle f:X\to Y} and g : Y → Z {\displaystyle g:Y\to Z} are local homeomorphisms then the composition g ∘ f : X → Z {\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 homeomorphism; explicitly, if f : X → Y {\displaystyle f:X\to Y} is a local homeomorphism then its restriction f | U : U → Y {\displaystyle f{\big \vert }_{U}:U\to Y} to any U {\displaystyle U} open subset of X {\displaystyle X} is also a local homeomorphism.

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

Preserved properties

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

  • X {\displaystyle X} is locally connected if and only if f ( X ) {\displaystyle f(X)} is;
  • X {\displaystyle X} is locally path-connected if and only if f ( X ) {\displaystyle f(X)} is;
  • X {\displaystyle X} is locally compact if and only if f ( X ) {\displaystyle f(X)} is;
  • X {\displaystyle X} is first-countable if and only if f ( X ) {\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.

Sheaves

The local homeomorphisms with codomain Y {\displaystyle Y} stand in a natural one-to-one correspondence with the sheaves of sets on Y ; {\displaystyle Y;} this correspondence is in fact an equivalence of categories. Furthermore, every continuous map with codomain Y {\displaystyle Y} gives rise to a uniquely defined local homeomorphism with codomain Y {\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.

See also

Notes

Citations