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The Sierpiński space is locally compact in senses (1), (2) and (3), and compact as well, but it is not Hausdorff or regular (or even preregular) so it is not locally compact in senses (4) or (5). The disjoint union of countably many copies of Sierpiński space is a non-compact space which is still locally compact in senses (1), (2) and (3 ...
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 .
For every topological space Y, the projection is a closed mapping [11] (see proper map). Every open cover linearly ordered by subset inclusion contains X. [12] Bourbaki defines a compact space (quasi-compact space) as a topological space where each filter has a cluster point (i.e., 8. in the above). [13]
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 ...
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 ...
The product of a CG-1 space and a locally compact space is CG-1. [27] (Here, locally compact is in the sense of condition (3) in the corresponding article, namely each point has a local base of compact neighborhoods.) The product of a CG-2 space and a locally compact Hausdorff space is CG-2. [28] [29]
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
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.