A continuous map between topological spaces is called a local homeomorphism if restricted to a neighbourhood of every point in its domain it becomes a homeomorphism.
One also says that this exhibits as an étale space over .
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 between topological spaces (a morphism in Top) such that
- for every , there is an open set such that the image is open in and the restriction of to is a homeomorphism ,
- for every , there is a neighbourhood of such that the image is a neighbourhood of and is a homeomorphism.
See also etale space.
For any topological space and for any set regarded as a discrete space, the projection
X \times S \to X
is a local homeomorphism.
For an open cover, let
X := \coprod_i U_i
be the disjoint union space of all the pathches. Equipped with the canonical projection
\coprod_i U_i \to Y
this is a local homeomorphism.
In general, for every sheaf of sets on ; there is a local homeomorphism such that over any open the set is naturally identified with the set of sections of . See étale space for more on this.
Revised on March 19, 2012 11:11:10
by Urs Schreiber