Properties
Every locally compact preregular space is, in fact, completely regular. It follows that every locally compact Hausdorff space is a Tychonoff space. Since straight regularity is a more familiar condition than either preregularity (which is usually weaker) or complete regularity (which is usually stronger), locally compact preregular spaces are normally referred to in the mathematical literature as locally compact regular spaces. Similarly locally compact Tychonoff spaces are usually just referred to as locally compact Hausdorff spaces.
Every locally compact Hausdorff space is a Baire space. That is, the conclusion of the Baire category theorem holds: the interior of every union of countably many nowhere dense subsets is empty.
A subspace X of a locally compact Hausdorff space Y is locally compact if and only if X can be written as the set-theoretic difference of two closed subsets of Y. As a corollary, a dense subspace X of a locally compact Hausdorff space Y is locally compact if and only if X is an open subset of Y. Furthermore, if a subspace X of any Hausdorff space Y is locally compact, then X still must be the difference of two closed subsets of Y, although the converse needn't hold in this case.
Quotient spaces of locally compact Hausdorff spaces are compactly generated. Conversely, every compactly generated Hausdorff space is a quotient of some locally compact Hausdorff space.
For locally compact spaces local uniform convergence is the same as compact convergence.
Read more about this topic: Locally Compact Space
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)