nLab local homeomorphism




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


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topological homotopy theory

Étale morphisms



A continuous map f:XYf : X \to Y between topological spaces is called a local homeomorphism if restricted to a neighbourhood of every point in its domain it becomes a homeomorphism onto its image which is required to be open.

One also says that this exhibits XX as an étale space over YY.

Notice that, despite the similarity of terms, local homeomorphisms are, in general, not local isomorphisms in any natural way. See the examples below.



A local homeomorphism is a continuous map p:EBp \colon E \to B between topological spaces (a morphism in Top) such that

  • for every element eEe \in E, there is an open subset UeU \ni e whose image p *(U)p_*(U) is open in BB and the restriction of pp to UU is a homeomorphism p| U:Up *(U)p|_U \colon U \to p_*(U),

or equivalently

  • for every eEe \in E, there is a neighbourhood UU of ee such that the image p *(U)p_*(U) is a neighbourhood of p(e)p(e) and p| U:Up *(U)p|_U: U \to p_*(U) is a homeomorphism.

See also at étale space.


For XX any topological space and for SS any set regarded as a discrete space, the projection

X×SX X \times S \to X

is a local homeomorphism.

For {U iY}\{U_i \to Y\} an open cover, let

X:= iU i X := \coprod_i U_i

be the disjoint union space of all the pathches. Equipped with the canonical projection

iU iY \coprod_i U_i \to Y

this is a local homeomorphism.

In general, for every sheaf AA of sets on YY; there is a local homeomorphism XYX \to Y such that over any open UXU \hookrightarrow X the set A(U)A(U) is naturally identified with the set of sections of YXY \to X. See étale space for more on this.



A local homeomorphism is an open map.


Let f:XYf \colon X \to Y be a local homeomorphism and UXU \subset X an open subset. We need to see that the image f(U)Yf(U) \subset Y is an open subset of YY. For this we may equivalently show that each yf(U)y \in f(U) has an open neighbourhood inside f(U)f(U).

But since any function is surjective onto its image, there exists xUx \in U with f(x)=yf(x) = y. By local homeomorphy of ff, this xXx \in X has an open neighbourhood U xXU_x \subset X with f |U x:U xf(U x)f_{|U_x} \colon U_x \to f(U_x) a homeomorphism. Since UU xU xU \cap U_x \subset U_x is an open neighbourhood of xx in U xU_x, the homeomorphy of f |U xf_{|U_x} implies that f(UU x)f(U)f(U \cap U_x) \subset f(U) is an open neighbourhood of f(x)=yf(x) = y.

Last revised on June 30, 2022 at 06:05:19. See the history of this page for a list of all contributions to it.