Editing Local homeomorphism (section)

==Properties==

A map is a local homeomorphism if and only if it is [[Continuous function (topology)|continuous]], [[Open map|open]], and [[Locally injective function|locally injective]]. In particular, every local homeomorphism is a continuous and [[open map]]. A [[bijective]] local homeomorphism is therefore a homeomorphism. 

Whether or not a function <math>f : X \to Y</math> is a local homeomorphism depends on its codomain. The [[Image (mathematics)|image]] <math>f(X)</math> of a local homeomorphism <math>f : X \to Y</math> is necessarily an open subset of its codomain <math>Y</math> and <math>f : X \to f(X)</math> will also be a local homeomorphism (that is, <math>f</math> will continue to be a local homeomorphism when it is considered as the surjective map <math>f : X \to f(X)</math> onto its image, where <math>f(X)</math> has the [[subspace topology]] inherited from <math>Y</math>). However, in general it is possible for <math>f : X \to f(X)</math> to be a local homeomorphism but <math>f : X \to Y</math> to {{em|not}} be a local homeomorphism (as is the case with the map <math>f : \R \to \R^2</math> defined by <math>f(x) = (x, 0),</math> for example). A map <math>f : X \to Y</math> is a local homomorphism if and only if <math>f : X \to f(X)</math> is a local homeomorphism and <math>f(X)</math> is an open subset of <math>Y.</math> 

Every [[Fiber (mathematics)|fiber]] of a local homeomorphism <math>f : X \to Y</math> is a [[Discrete space|discrete subspace]] of its [[Domain of a function|domain]] <math>X.</math>

A local homeomorphism <math>f : X \to Y</math> transfers "local" topological properties in both directions: 
* <math>X</math> is [[Locally connected space|locally connected]] if and only if <math>f(X)</math> is;
* <math>X</math> is [[locally path-connected]] if and only if <math>f(X)</math> is;
* <math>X</math> is [[Locally compact space|locally compact]] if and only if <math>f(X)</math> is;
* <math>X</math> is [[First-countable space|first-countable]] if and only if <math>f(X)</math> 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]] <math>Y</math> stand in a natural one-to-one correspondence with the [[Sheaf (mathematics)|sheaves]] of sets on <math>Y;</math> this correspondence is in fact an [[equivalence of categories]]. Furthermore, every continuous map with codomain <math>Y</math> gives rise to a uniquely defined local homeomorphism with codomain <math>Y</math> in a natural way. All of this is explained in detail in the article on [[Sheaf (mathematics)|sheaves]].