). However, in general it is possible for to be a local homeomorphism but to be a local homeomorphism (as is the case with the map defined by for example). A map is a local homomorphism if and only if is a local homeomorphism and is an open subset of Every fiber of a local homeomorphism is a discrete subspace of its domain A local homeomorphism transfers "local" topological properties in both directions: is locally connected if and only if is; is locally path-connected if and only if is; is locally compact if and only if is; is first-countable if and only if 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 with codomain stand in a natural one-to-one correspondence with the sheaves of sets on this correspondence is in fact an equivalence of categories. Furthermore, every continuous map with codomain gives rise to a uniquely defined local homeomorphism with codomain in a natural way. All of this is explained in detail in the article on 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 manifolds, we obtain the local diffeomorphisms; for schemes, we have the formally étale morphisms and the étale morphisms; and for toposes, we get the étale geometric morphisms. See also Notes Citations References Theory of continuous functions Functions and mappings General topology