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In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood .
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.
A Hausdorff, Baire space that is also σ-compact, must be locally compact at at least one point. If G is a topological group and G is locally compact at one point, then G is locally compact everywhere. Therefore, the previous property tells us that if G is a σ-compact, Hausdorff topological group that is also a Baire space, then G is
By homogeneity, local compactness of the underlying space for a topological group need only be checked at the identity. That is, a group G is a locally compact space if and only if the identity element has a compact neighborhood.
The space of ordinals at most equal to Ω, the first uncountable ordinal with the order topology is a compact topological space. The measure which equals 1 on any Borel set that contains an uncountable closed subset of [1, Ω) , and 0 otherwise, is Borel but not Radon, as the one-point set {Ω} has measure zero but any open neighbourhood of it ...
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 contain X itself.
In topology, a topological space is called a compactly generated space or k-space if its topology is determined by compact spaces in a manner made precise below. There is in fact no commonly agreed upon definition for such spaces, as different authors use variations of the definition that are not exactly equivalent to each other.
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 .