Local homeomorphism

2025-10-20T01:02:27Z

2601:447:CD7E:1870:1141:B2A9:3717:8856: Without these braces, there is not proper horizontal spacing in the typography.

{{Short description|Mathematical function revertible near each point}}
In [[mathematics]], more specifically [[topology]], a '''local homeomorphism''' is a [[Function (mathematics)|function]] between [[topological space]]s that, intuitively, preserves local (though not necessarily global) structure.
If <math>f : X \to Y</math> is a local homeomorphism, <math>X</math> is said to be an '''étale space''' over <math>Y.</math> Local homeomorphisms are used in the study of [[Sheaf (mathematics)|sheaves]]. Typical examples of local homeomorphisms are [[covering map]]s.

A topological space <math>X</math> is '''locally homeomorphic''' to <math>Y</math> if every point of <math>X</math> has a neighborhood that is [[homeomorphic]] to an open subset of <math>Y.</math>
For example, a [[manifold]] of dimension <math>n</math> is locally homeomorphic to <math>\R^n.</math>

If there is a local homeomorphism from <math>X</math> to <math>Y,</math> then <math>X</math> is locally homeomorphic to <math>Y,</math> but the converse is not always true.
For example, the two dimensional [[sphere]], being a manifold, is locally homeomorphic to the plane <math>\R^2,</math> but there is no local homeomorphism <math>S^2 \to \R^2.</math>

==Formal definition==

A function <math>f : X \to Y</math> between two [[topological space]]s is called a {{em|local homeomorphism}}<ref>{{cite book |last=Munkres|first=James R.|authorlink=James Munkres|title=Topology|edition=2nd|publisher=[[Prentice Hall]]|year=2000|isbn=0-13-181629-2}}</ref> if every point <math>x \in X</math> has an [[open neighborhood]] <math>U</math> whose [[Image (mathematics)|image]] <math>f(U)</math> is open in <math>Y</math> and the [[Restriction (mathematics)|restriction]] <math>f\big\vert_U : U \to f(U)</math> is a [[homeomorphism]] (where the respective [[Subspace topology|subspace topologies]] are used on <math>U</math> and on <math>f(U)</math>).

==Examples and sufficient conditions==

'''Covering maps'''

Every homeomorphism is a local homeomorphism. The function <math>\R \to S^1</math> defined by <math>t \mapsto e^{it}</math> (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 <math>f : S^1 \to S^1</math> defined by <math>f(z) = z^n,</math> where <math>n</math> is a fixed integer, wraps the circle around itself <math>n</math> times (that is, has [[winding number]] <math>n</math>) and is a local homeomorphism for all non-zero <math>n,</math> but it is a homeomorphism only when it is [[bijective]] (that is, only when <math>n = 1</math> or <math>n = -1</math>).

Generalizing the previous two examples, every [[Covering space|covering map]] is a local homeomorphism; in particular, the [[universal cover]] <math>p : C \to Y</math> of a space <math>Y</math> is a local homeomorphism.
In certain situations the converse is true. For example: if <math>p : X \to Y</math> is a [[Proper map|proper]] local homeomorphism between two [[Hausdorff space]]s and if <math>Y</math> is also [[locally compact]], then <math>p</math> is a covering map.

'''Inclusion maps of open subsets'''

If <math>U \subseteq X</math> is any subspace (where as usual, <math>U</math> is equipped with the [[Subspace (topology)|subspace topology]] induced by <math>X</math>) then the [[inclusion map]] <math>i : U \to X</math> is always a [[topological embedding]]. It is a local homeomorphism if and only if <math>U</math> is open in <math>X.</math>

'''Invariance of domain'''

[[Invariance of domain]] guarantees that if <math>f : U \to \R^n</math> is a [[Continuous function|continuous]] [[injective map]] from an open subset <math>U</math> of <math>\R^n,</math> then <math>f(U)</math> is open in <math>\R^n</math> and <math>f : U \to f(U)</math> is a [[homeomorphism]].
Consequently, a continuous map <math>f : U \to \R^n</math> from an open subset <math>U \subseteq \R^n</math> will be a local homeomorphism if and only if it is a [[Locally injective function|''locally'' injective map]] (meaning that every point in <math>U</math> has a [[Neighborhood (topology)|neighborhood]] <math>N</math> such that the restriction of <math>f</math> to <math>N</math> is injective).

'''Local homeomorphisms in analysis'''

It is shown in [[complex analysis]] that a complex [[Holomorphic function|analytic]] function <math>f : U \to \Complex</math> (where <math>U</math> is an open subset of the [[complex plane]] <math>\Complex</math>) is a local homeomorphism precisely when the [[derivative]] <math>f^{\prime}(z)</math> is non-zero for all <math>z \in U.</math>
The function <math>f(z) = z^n</math>, with fixed integer <math>n</math>, defined on an open disk around <math>0</math>, is not a local homeomorphism when <math>n \geq 2.</math>
In that case <math>0</math> is a point of "[[Ramification (mathematics)|ramification]]" (intuitively, <math>n</math> sheets come together there).

Using the [[inverse function theorem]] one can show that a continuously differentiable function <math>f : U \to \R^n</math> (where <math>U</math> is an open subset of <math>\R^n</math>) is a local homeomorphism if the derivative <math>D_x f</math> is an invertible linear map (invertible square matrix) for every <math>x \in U.</math>
(The converse is false, as shown by the local homeomorphism <math>f : \R \to \R</math> with <math>f(x) = x^3</math>).
An analogous condition can be formulated for maps between [[differentiable manifold]]s.

'''Local homeomorphisms and Hausdorffness'''

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

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

'''Local homeomorphisms and fibers'''

Suppose <math>f : X \to Y</math> is a continuous [[Open map|open]] surjection between two [[Hausdorff space|Hausdorff]] [[Second-countable space|second-countable]] spaces where <math>X</math> is a [[Baire space]] and <math>Y</math> is a [[normal space]]. If every [[Fiber (mathematics)|fiber]] of <math>f</math> is a [[Discrete space|discrete subspace]] of <math>X</math> (which is a necessary condition for <math>f : X \to Y</math> to be a local homeomorphism) then <math>f</math> is a <math>Y</math>-valued local homeomorphism on a dense open subset of <math>X.</math>
To clarify this statement's conclusion, let <math>O = O_f</math> be the (unique) largest open subset of <math>X</math> such that <math>f\big\vert_O : O \to Y</math> is a local homeomorphism.<ref group="note">The assumptions that <math>f</math> is continuous and open imply that the set <math>O = O_f</math> is equal to the union of all open subsets <math>U</math> of <math>X</math> such that the restriction <math>f\big\vert_U : U \to Y</math> is an [[Injective function|injective map]].</ref>
If every [[Fiber (mathematics)|fiber]] of <math>f</math> is a [[Discrete space|discrete subspace]] of <math>X</math> then this open set <math>O</math> is necessarily a [[Dense subset|{{em|dense}} subset]] of <math>X.</math>
In particular, if <math>X \neq \varnothing</math> then <math>O \neq \varnothing;</math> a conclusion that may be false without the assumption that <math>f</math>'s fibers are discrete (see this footnote<ref group="note">Consider the continuous open surjection <math>f : \R \times \R \to \R</math> defined by <math>f(x, y) = x.</math> The set <math>O = O_f</math> for this map is the empty set; that is, there does not exist any non-empty open subset <math>U</math> of <math>\R \times \R</math> for which the restriction <math>f\big\vert_U : U \to \R</math> is an injective map.</ref> for an example).
One corollary is that every continuous open surjection <math>f</math> between [[Completely metrizable space|completely metrizable]] second-countable spaces that has [[Discrete space|discrete]] fibers is "almost everywhere" a local homeomorphism (in the topological sense that <math>O_f</math> is a dense open subset of its domain).
For example, the map <math>f : \R \to [0, \infty)</math> defined by the polynomial <math>f(x) = x^2</math> is a continuous open surjection with discrete fibers so this result guarantees that the maximal open subset <math>O_f</math> is dense in <math>\R;</math> with additional effort (using the [[inverse function theorem]] for instance), it can be shown that <math>O_f = \R \setminus \{0\},</math> which confirms that this set is indeed dense in <math>\R.</math> This example also shows that it is possible for <math>O_f</math> to be a {{em|proper}} dense subset of <math>f</math>'s domain.
Because [[Fundamental theorem of algebra|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.<ref group="note">And even if the polynomial function is not an open map, then this theorem may nevertheless still be applied (possibly multiple times) to restrictions of the function to appropriately chosen subsets of the domain (based on consideration of the map's local minimums/maximums).</ref>

==Properties==

A map <math>f : X \to Y</math> is a local homeomorphism if and only if it is [[Continuous function (topology)|continuous]], [[Open map|open]], and [[Locally injective function|locally injective]] (the latter means that every point in <math>X</math> has a [[Neighborhood (topology)|neighborhood]] <math>N</math> such that the restriction of <math>f</math> to <math>N</math> is injective). It follows that the map <math>f</math> is a homeomorphism if and only if it is a [[bijective]] local homeomorphism.

Every [[Fiber (mathematics)|fiber]] of a local homeomorphism <math>f : X \to Y</math> is a [[Discrete space|discrete subspace]] of its [[Domain of a function|domain]] <math>X.</math>

Whether or not a function is a local homeomorphism depends on its codomain: A map <math>f : X \to Y</math> is a local homomorphism if and only if the surjection <math>f : X \to f(X)</math> is a local homeomorphism (where <math>f(X)</math> has the [[subspace topology]] inherited from <math>Y</math>) and <math>f(X)</math> is an open subset of <math>Y.</math>

=== Local homeomorphisms and composition of functions ===
The [[Function composition|composition]] of two local homeomorphisms is a local homeomorphism; explicitly, if <math>f : X \to Y</math> and <math>g : Y \to Z</math> are local homeomorphisms then the composition <math>g \circ f : X \to Z</math> 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 <math>f : X \to Y</math> is a local homeomorphism then its restriction <math>f\big\vert_U : U \to Y</math> to any <math>U</math> open subset of <math>X</math> is also a local homeomorphism.

If <math>f : X \to Y</math> is continuous while both <math>g : Y \to Z</math> and <math>g \circ f : X \to Z</math> are local homeomorphisms, then <math>f</math> is also a local homeomorphism.

=== Preserved properties ===
A local homeomorphism <math>f : X \to Y</math> transfers "local" topological properties in both directions:
* <math>X</math> is [[Locally connected space|locally connected]] if and only if <math>f(X)</math> is;
* <math>X</math> is [[locally path-connected]] if and only if <math>f(X)</math> is;
* <math>X</math> is [[Locally compact space|locally compact]] if and only if <math>f(X)</math> is;
* <math>X</math> is [[First-countable space|first-countable]] if and only if <math>f(X)</math> 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]] <math>Y</math> stand in a natural one-to-one correspondence with the [[Sheaf (mathematics)|sheaves]] of sets on <math>Y;</math> this correspondence is in fact an [[equivalence of categories]]. Furthermore, every continuous map with codomain <math>Y</math> gives rise to a uniquely defined local homeomorphism with codomain <math>Y</math> in a natural way. All of this is explained in detail in the article on [[Sheaf (mathematics)|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 manifold]]s, we obtain the [[local diffeomorphism]]s; for [[Scheme (mathematics)|schemes]], we have the [[formally étale morphism]]s and the [[étale morphism]]s; and for [[topos]]es, we get the [[étale geometric morphism]]s.

==See also==

* {{annotated link|Diffeomorphism}}
* {{annotated link|Homeomorphism}}
* {{annotated link|Isomorphism}}
* {{annotated link|Invariance of domain}}
* {{annotated link|Local diffeomorphism}}
* {{annotated link|Locally Hausdorff space}}
* {{annotated link|Non-Hausdorff manifold}}

==Notes==

{{reflist|group=note}}

==Citations==

{{reflist}}

==References==

* {{Bourbaki General Topology Part I Chapters 1-4}} 
* {{Munkres Topology|edition=2}} 
* {{Willard General Topology}} 

{{Topology}}
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[[Category:Theory of continuous functions]]
[[Category:Functions and mappings]]
[[Category:General topology]]
[[Category:Homeomorphisms| ]]

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Local homeomorphism