CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A continuous map $f : X \to Y$ between topological spaces is called a local homeomorphism if restricted to a neighbourhood of every point in its domain it becomes a homeomorphism.
One also says that this exhibits $X$ as an étale space over $Y$.
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 $p : E \to B$ between topological spaces (a morphism in Top) such that
or equivalently
See also etale space.
For $Y$ any topological space and for $S$ any set regarded as a discrete space, the projection
is a local homeomorphism.
For $\{U_i \to Y\}$ an open cover, let
be the disjoint union space of all the pathches. Equipped with the canonical projection
this is a local homeomorphism.
In general, for every sheaf $A$ of sets on $Y$; there is a local homeomorphism $X \to Y$ such that over any open $U \hookrightarrow X$ the set $A(U)$ is naturally identified with the set of sections of $Y \to X$. See étale space for more on this.