ENCYCLOPEDIA 4U .com

 Web Encyclopedia4u.com

# Local homeomorphism

In topology, a local homeomorphism is a map f from one topological space X to another, Y, that respects locally the topological structure. More precisely, for all points x of X there should be an open neighbourhood N of x, such that f(N) is open in Y and f restricted to N is a homeomorphism from N to f(N).

### Some examples

Taking X and Y to be the circle S1, regarded as the quotient space R/Z, we can take f to be the function induced by multiplication by n for any integer n. Then this is a local homeomorphism for all non-zero n, but a homeomorphism only in the cases where it is bijective, i.e. n = 1 and -1.

It is shown in complex analysis that a complex analytic function f gives a local homeomorphism precisely when the derivative f'(z) is non-zero for all z in the domain of f. The function f(z) = zn on an open disk round 0 is not a local homeomorphism at 0 when n is at least 2. In that case 0 is a point of "ramification" (intuitively, n sheets come together there).

All covering maps are local homeomorphisms; in particular, the universal cover p : CX of a space X is a local homeomorphism.

Every homeomorphism is of course also a local homeomorphism.

### Properties

Every local homeomorphism is a continuous and open map. A local homeomorphism f : XY preserves "local" topological properties:

Content on this web site is provided for informational purposes only. We accept no responsibility for any loss, injury or inconvenience sustained by any person resulting from information published on this site. We encourage you to verify any critical information with the relevant authorities.