local homeomorphism

Every local homeomorphism is a continuous and open map.

By definition, every homeomorphism is also a local homeomorphism.

If is a local homeomorphism, then is quasi-open.

Because a sheaf (thought of as an étalé space) can be considered a local homeomorphism, the notions were closely interlinked at the time.

A bijective local homeomorphism is therefore a homeomorphism.

A covering space is a fiber bundle such that the bundle projection is a local homeomorphism.

Openness is essential here: the inclusion map of a non-open subset of Y never yields a local homeomorphism.

A local homeomorphism f : X Y preserves "local" topological properties:

In this case, C is called a covering space of X. The definition implies that every covering map is a local homeomorphism.

In mathematics, more specifically topology, a local homeomorphism is intuitively a function, f, between topological spaces that preserves local structure.

At most points of the Riemann sphere, this transformation is a local homeomorphism: it maps a small disk centered at any point in a one-to-one way into another disk.

Further, because a covering map is a local homeomorphism (in this case a local isometry), both the spherical and the corresponding projective polyhedra have the same abstract vertex figure.

If U is an open subset of Y equipped with the subspace topology, then the inclusion map i : U Y is a local homeomorphism.

Every covering map is a local homeomorphism; in particular, the universal cover p : C Y of a space Y is a local homeomorphism.

Note that it is not a covering map - while it is a local homeomorphism near the origin, it is not a covering map at rotations by 180 degrees.

Depending on the context, we can take this as local homeomorphism for the strong topology, over the complex numbers, or as an étale morphism in general (under some slightly stronger hypotheses, on flatness and separability).

In certain situations, the converse is true: if X and Y are locally compact spaces and p : X Y is a proper local homeomorphism, then p is a covering map.

A function is a local homeomorphism if for every point x in X there exists an open set U containing x, such that the image is open in Y and the restriction is a homeomorphism.

That is, if X is locally Euclidean of dimension n and f : Y X is a local homeomorphism, then Y is locally Euclidean of dimension n. In particular, being locally Euclidean is a topological property.

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.