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Every locally compact regular space, in particular every locally compact Hausdorff space, is a Baire space. [14] [15] That is, the conclusion of the Baire category theorem holds: the interior of every countable union of nowhere dense subsets is empty. A subspace X of a locally compact Hausdorff space Y is locally compact if and only if X is ...
Every topological space X is an open dense subspace of a compact space having at most one point more than X, by the Alexandroff one-point compactification. By the same construction, every locally compact Hausdorff space X is an open dense subspace of a compact Hausdorff space having at most one point more than X.
In algebra, a locally compact field is a topological field whose topology forms a locally compact Hausdorff space. [1] These kinds of fields were originally introduced in p-adic analysis since the fields are locally compact topological spaces constructed from the norm | | on .
Every quotient of a locally compact group is locally compact. The product of a family of locally compact groups is locally compact if and only if all but a finite number of factors are actually compact. Topological groups are always completely regular as topological spaces. Locally compact groups have the stronger property of being normal.
Given some notion of equivalence (e.g., homeomorphism, diffeomorphism, isometry) between topological spaces, two spaces are said to be locally equivalent if every point of the first space has a neighborhood which is equivalent to a neighborhood of the second space. For instance, the circle and the line are very different objects. One cannot ...
The theory of Radon measures has most of the good properties of the usual theory for locally compact spaces, but applies to all Hausdorff topological spaces. The idea of the definition of a Radon measure is to find some properties that characterize the measures on locally compact spaces corresponding to positive functionals, and use these ...
The first definition is usually taken for locally compact, countably compact, metrizable, separable, countable; the second for locally connected. [15] Locally closed subset A subset of a topological space that is the intersection of an open and a closed subset. Equivalently, it is a relatively open subset of its closure. Locally compact
Paracompact Hausdorff spaces are normal. Locally compact. A space is locally compact if every point has a local base consisting of compact neighbourhoods. Slightly different definitions are also used. Locally compact Hausdorff spaces are always Tychonoff. Ultraconnected compact. In an ultra-connected compact space X every open cover must ...