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.
Read more about this topic: Local Homeomorphism
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)