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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 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.
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 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 ...
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]
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
If G is a locally compact Hausdorff group, G carries an essentially unique left-invariant countably additive Borel measure μ called a Haar measure.Using the Haar measure, one can define a convolution operation on the space C c (G) of complex-valued continuous functions on G with compact support; C c (G) can then be given any of various norms and the completion will be a group algebra.