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
or:
$G$ is a locally compact topological groups
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)
A common special case of either condition in Prop. is:
$G$ is a compact Lie group
and $H$ is a closed subgroup.
(This is a special case of Gleason’s condition since Lie groups are Hausdorff spaces and closed subspaces of compact Hausdorff spaces are equivalently compact subspaces.)
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)
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