Local homeomorphism

Local homeomorphism

In topology, a local homeomorphism is a map from one topological space to another that respects locally the topological structure of the two spaces. More precisely, a continuous map "f" : "X" → "Y" is a local homeomorphism if for every point "x" of "X" there exists an open neighbourhood "U" of "x" such that "f"("U") is open in "Y" and "f"|"U" : "U" → "f"("U") is a homeomorphism.

Some examples

Every homeomorphism is, of course, also a local homeomorphism.

If "U" is an open subset of "Y" equipped with the subspace topology, then the inclusion map "i" : "U" → "Y" is a local homeomorphism. Openness is essential here: the inclusion map of a non-open subset of "Y" never yields a local homeomorphism.

Let "f" : "S"1 → "S"1 be the map that wraps the circle around itself "n" times (i.e. has winding number "n"). 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 or -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") = "z""n" on an open disk around 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.

Properties

Every local homeomorphism is a continuous and open map. A bijective local homeomorphism is therefore a homeomorphism.

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

If "f" : "X" → "Y" is a local homeomorphism and "U" is an open subset of "X", then the restriction "f"|"U" is also a local homeomorphism.

If "f" : "X" → "Y" and "g" : "Y" → "Z" are local homeomorphisms, then the composition "gf" : "X" → "Z" is also a local homeomorphism.

The local homeomorphisms with codomain "Y" stand in a natural 1-1 correspondence with the sheaves of sets on "Y". Furthermore, every continuous map with codomain "Y" gives rise to a uniquely defined local homeomorphism with codomain "Y" in a natural way. All of this is explained in detail in the article on sheaves.

ee also

*Local diffeomorphism


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