topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Related concepts
A priori a locally compact topological group is a topological group whose underlying topological space is locally compact.
Typically it is also assumed that is Hausdorff. (Notice that if not, then is Hausdorff.).
One often says just “locally compact group”.
We take here locally compact groups to be also Hausdorff.
Locally compact topological groups are the standard object of study in classical abstract harmonic analysis. The crucial properties of locally compact groups is that they posses a left (right) Haar measure and that has a structure of a Banach -algebra.
A left (right) Haar measure on a locally compact topological group is a nonzero Radon measure which is invariant under the left (right) multiplications by elements in the group. A topological subgroup of a locally compact topological group is itself locally compact (in induced topology) iff it is closed in .
Again taking locally compact groups to be Hausdorff, such are complete both with respect to their left uniformity and their right uniformity. For if is a Cauchy net in and is a compact neighborhood of the identity , then there is so large that for all . Those elements converge to a point since is compact, and the original net converges to . A similar argument is used for the right uniformity.
See at Pontrjagin duality.
Linus Kramer, Locally Compact Groups, 2017 (pdf, pdf)
Gerald B. Folland, A course in abstract harmonic analysis, Studies in Adv. Math. CRC Press 1995
See also:
Last revised on August 21, 2021 at 19:17:52. See the history of this page for a list of all contributions to it.