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
Given a topological group $G$ and a subgroup $H \subset G$ it is often oft interest to known that the coset space coprojection $G \to G/H$ admits local sections. For instance, these yield canonical examples of $H$-principal bundles, of H-structures and of equivariant bundles.
(sufficient conditions for coset space coprojections having local sections)
Let $G$ be a topological group and $H \subset G$ a subgroup.
Then sufficient conditions for the coset space coprojection $G \overset{q}{\to} G/H$ to admit local sections, in that there is an open cover $\underset{i \in I}{\sqcup}U_i \to G/H$ and a continuous section $\sigma_{\mathcal{U}}$ of the pullback of $q$ to the cover,
include the following:
$G$ is any topological group
and $H$ is a compact Lie group
$G$ is a locally compact topological group
which is moreover a separable metric space of finite dimension
and $H \subset G$ is a closed subgroup.
(Mostert 53, Theorem 3, see also Karube 58, Theorem 2)
$G$ is a (finite-dimensional) Lie group
and $H$ is a closed subgroup
For $\mathcal{H}$ an infinite-dimensional separable complex Hilbert space, the coset space of the unitary group U(ℋ) by its circle subgroup U(1) is the projective unitary group $\mathrm{U}(\mathcal{H})/U(1) \; \simeq$ PU(ℋ), both in their weak/strong operator topology. This coset space coprojection falls outside the applicatibility of Prop. , and yet it does admit local sections (Simms 1970, Thm. 1).
Examples of quotient coprojections $G \to G/H$ without local sections are given in Karube 58, Sec. 3.
Hans Samelson, Beiträge zur Topologie der Gruppenmannigfaltigkeiten, Ann. of Math. 2, 42, (1941), 1091 - 1137. (jstor:1970463, doi:10.2307/1970463)
Andrew Gleason, Spaces With a Compact Lie Group of Transformations, Proceedings of the American Mathematical Society Vol. 1, No. 1 (Feb., 1950), pp. 35-43 (jstor:2032430, doi:10.2307/2032430)
Paul Mostert, Local Cross Sections in Locally Compact Groups, Proceedings of the American Mathematical Society, Vol. 4, No. 4 (Aug., 1953), pp.645-649 (jstor:2032540, doi:10.2307/2032540)
Takashi Karube, On the local cross-sections in locally compact groups, J. Math. Soc. Japan 10(4): 343-347 (October, 1958) (doi:10.2969/jmsj/01040343)
Tammo tom Dieck, Theodor Bröcker, Thm. 4.3 on p. 33 of: Representations of compact Lie groups, Springer 1985 (doi:10.1007/978-3-662-12918-0)
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