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The first two examples show that a subset of a locally compact space need not be locally compact, which contrasts with the open and closed subsets in the previous section. The last example contrasts with the Euclidean spaces in the previous section; to be more specific, a Hausdorff topological vector space is locally compact if and only if it ...
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 mathematics, a locally compact group is a topological group G for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are locally compact and such groups have a natural measure called the Haar measure .
More precisely, let X be a topological space. Then the Alexandroff extension of X is a certain compact space X* together with an open embedding c : X → X* such that the complement of X in X* consists of a single point, typically denoted ∞. The map c is a Hausdorff compactification if and only if X is a locally compact, noncompact Hausdorff ...
In mathematics, in general topology, compactification is the process or result of making a topological space into a compact space. [1] A compact space is a space in which every open cover of the space contains a finite subcover. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by ...
In fact, every CG-2 space is a quotient space of a CG-3 space (namely, some locally compact Hausdorff space); but there are CG-2 spaces that are not CG-3. For a concrete example, the Sierpiński space is not CG-3, but is homeomorphic to the quotient of the compact interval [ 0 , 1 ] {\displaystyle [0,1]} obtained by identifying ( 0 , 1 ...
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 Q p {\displaystyle \mathbb {Q} _{p}} are locally compact topological spaces constructed from the norm | ⋅ | p {\displaystyle |\cdot |_{p ...
A topological group is called locally compact if the underlying topological space is locally compact and Hausdorff; the topological group is called abelian if the underlying group is abelian. Examples of locally compact abelian groups include: for n a positive integer, with vector addition as group operation.