Locally compact space
topological space such that every point has a neighbourhood with compact closure
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.
When locally compact spaces are Hausdorff they are called locally compact Hausdorff, which are of particular interest in mathematical analysis.
Formal definition
Let X be a topological space. Most commonly X is called locally compact if every point x of X has a compact neighbourhood, i.e., there exists an open set U and a compact set K, such that
x
∈
U
⊆
K
{\displaystyle x\in U\subseteq K}
.
There are other common definitions, which are all equivalent if X is a Hausdorff space (or preregular), but are not equivalent in general:
1. every point of X has a compact neighbourhood.
2. every point of X has a closed compact neighbourhood.
2′.
Begin with the source’s own compact description: “Locally compact space” is topological space such that every point has a neighbourhood with compact closure. The dossier treats that line as a proposition to test through Locally, compact and space, not as a finished interpretation.
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This entry incorporates text from “Locally compact space” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.