Inmathematics, more specificallytopology, alocal homeomorphism is afunction betweentopological spaces that, intuitively, preserves local (though not necessarily global) structure. If${\displaystyle f:X\to Y}$ is a local homeomorphism,${\displaystyle X}$ is said to be anétale space over${\displaystyle Y.}$ Local homeomorphisms are used in the study ofsheaves. Typical examples of local homeomorphisms arecovering maps.

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

If there is a local homeomorphism from${\displaystyle X}$ to${\displaystyle Y,}$ then${\displaystyle X}$ is locally homeomorphic to${\displaystyle Y,}$ but the converse is not always true. For example, the two dimensionalsphere, being a manifold, is locally homeomorphic to the plane${\displaystyle {\mathbb{R}}^2,}$ but there is no local homeomorphism${\displaystyle S^2\to {\mathbb{R}}^2.}$

A function${\displaystyle f:X\to Y}$ between twotopological spaces is called alocal homeomorphism^[1] if every point${\displaystyle x\in X}$ has anopen neighborhood${\displaystyle U}$ whoseimage${\displaystyle f(U)}$ is open in${\displaystyle Y}$ and therestriction${\displaystyle f{|}_U:U\to f(U)}$ is ahomeomorphism (where the respectivesubspace topologies are used on${\displaystyle U}$ and on${\displaystyle f(U)}$).

Local homeomorphisms versus homeomorphisms

Every homeomorphism is a local homeomorphism. But a local homeomorphism is a homeomorphism if and only if it isbijective. 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 thereal line around thecircle) 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, haswinding number${\displaystyle n}$), is a local homeomorphism for all non-zero${\displaystyle n,}$ but it is a homeomorphism only when it isbijective (that is, only when${\displaystyle n=1}$ or${\displaystyle n=-1}$).

Generalizing the previous two examples, everycovering map is a local homeomorphism; in particular, theuniversal 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 aproper local homeomorphism between twoHausdorff spaces and if${\displaystyle Y}$ is alsolocally compact, then${\displaystyle p}$ is a covering map.

Local homeomorphisms and composition of functions

Thecomposition 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 thesubspace topology induced by${\displaystyle X}$) then theinclusion map${\displaystyle i:U\to X}$ is always atopological 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 acontinuousinjective 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 ahomeomorphism. 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 alocally injective map (meaning that every point in${\displaystyle U}$ has aneighborhood${\displaystyle N}$ such that the restriction of${\displaystyle f}$ to${\displaystyle N}$ is injective).

Local homeomorphisms in analysis

It is shown incomplex analysis that a complexanalytic function${\displaystyle f:U\to \mathbb{C}}$ (where${\displaystyle U}$ is an open subset of thecomplex plane${\displaystyle \mathbb{C}}$) is a local homeomorphism precisely when thederivative${\displaystyle f^{\mathrm{\prime }}(z)}$ 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 theinverse 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 betweendifferentiable manifolds.

Local homeomorphisms and fibers

Suppose${\displaystyle f:X\to Y}$ is a continuousopen surjection between twoHausdorffsecond-countable spaces where${\displaystyle X}$ is aBaire space and${\displaystyle Y}$ is anormal space. If everyfiber of${\displaystyle f}$ is adiscrete 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.^{[note 1]} If everyfiber of${\displaystyle f}$ is adiscrete subspace of${\displaystyle X}$ then this open set${\displaystyle O}$ is necessarily adense 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^{[note 2]} for an example). One corollary is that every continuous open surjection${\displaystyle f}$ betweencompletely metrizable second-countable spaces that hasdiscrete 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 theinverse 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 aproper dense subset of${\displaystyle f}$'s domain. Becauseevery 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.^{[note 3]}

Local homeomorphisms and Hausdorffness

There exist local homeomorphisms${\displaystyle f:X\to Y}$ where${\displaystyle Y}$ is aHausdorff space but${\displaystyle X}$ is not. Consider for instance thequotient space${\displaystyle X=\left(\mathbb{R}\bigsqcup \mathbb{R}\right)/\sim ,}$ where theequivalence relation${\displaystyle \sim }$ on thedisjoint 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 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.

A map is a local homeomorphism if and only if it iscontinuous,open, andlocally injective. In particular, every local homeomorphism is a continuous andopen map. Abijective local homeomorphism is therefore a homeomorphism.

Whether or not a function${\displaystyle f:X\to Y}$ is a local homeomorphism depends on its codomain. Theimage${\displaystyle f(X)}$ 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 thesubspace topology inherited from${\displaystyle Y}$). However, in general it is possible for${\displaystyle f:X\to f(X)}$ to be a local homeomorphism but${\displaystyle f:X\to Y}$ tonot 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${\displaystyle Y.}$

Everyfiber of a local homeomorphism${\displaystyle f:X\to Y}$ is adiscrete subspace of itsdomain${\displaystyle X.}$

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

• ${\displaystyle X}$ islocally connected if and only if${\displaystyle f(X)}$ is;
• ${\displaystyle X}$ islocally path-connected if and only if${\displaystyle f(X)}$ is;
• ${\displaystyle X}$ islocally compact if and only if${\displaystyle f(X)}$ is;
• ${\displaystyle X}$ isfirst-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 withcodomain${\displaystyle Y}$ stand in a natural one-to-one correspondence with thesheaves of sets on${\displaystyle Y;}$ this correspondence is in fact anequivalence 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 onsheaves.

The idea of a local homeomorphism can be formulated in geometric settings different from that of topological spaces. Fordifferentiable manifolds, we obtain thelocal diffeomorphisms; forschemes, we have theformally étale morphisms and theétale morphisms; and fortoposes, we get theétale geometric morphisms.

• Diffeomorphism – Isomorphism of differentiable manifolds
• Homeomorphism – Mapping which preserves all topological properties of a given space
• Isomorphism – In mathematics, invertible homomorphism
• Invariance of domain – Theorem in topology about homeomorphic subsets of Euclidean space
• Local diffeomorphism
• Locally Hausdorff space – Space such that every point has a Hausdorff neighborhood
• Non-Hausdorff manifold – generalization of manifoldsPages displaying wikidata descriptions as a fallback

• Bourbaki, Nicolas (1989) [1966].General Topology: Chapters 1–4 [Topologie Générale].Éléments de mathématique. Berlin New York: Springer Science & Business Media.ISBN 978-3-540-64241-1.OCLC 18588129.
• Munkres, James R. (2000).Topology (2nd ed.).Upper Saddle River, NJ:Prentice Hall, Inc.ISBN 978-0-13-181629-9.OCLC 42683260.(accessible to patrons with print disabilities)
• Willard, Stephen (2004) [1970].General Topology.Mineola, N.Y.:Dover Publications.ISBN 978-0-486-43479-7.OCLC 115240.

• Theory of continuous functions
• Functions and mappings
• General topology
• Homeomorphisms

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