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 : C → X 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 : X → Y preserves "local" topological properties:

• X is locally connected if and only if f(X) is
• X is locally path-connected if and only if f(X) is
• X is locally compact if and only if f(X) is
• X is first countable if and only if f(X) is