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
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
A pointed topological space $(X,x)$ is called well-pointed if the base-point inclusion $\{x\} \xhookrightarrow{\;} X$ is a closed Hurewicz cofibration (e.g. tom Dieck 2008, p. 102). If the topological space is Hausdorff, then closedness is implied (by this Prop.) and one may require just a Hurewicz cofibration (eg. Bredon 1993, VII, Def. 1.8).
A topological group is called well-pointed if it is so at its neutral element, hence if $\{\mathrm{e}\} \xhookrightarrow{\;} G$ is a closed Hurewicz cofibration.
A simplicial topological group is well-pointed if all its component groups are.
A key property of well-pointed topological groups (in the convenient context of compactly generated weak Hausdorff spaces) is that the nerves of their delooping groupoids are good simplicial topological spaces (by this Ex.). Similarly, the underlying simplicial topological spaces of well-pointed simplicial topological groups are good (this Prop.). These facts explain the key role of well-pointedness in classifying space-theory, where it ensures that plain topological realization of simplicial topological spaces coincides, up to weak homotopy equivalence, with fat geometric realization and hence with the homotopy colimits.
Every locally Euclidean Hausdorff space is well-pointed, in particular every topological manifold is well pointed. In fact, every paracompact Banach manifold is well-pointed.
(the projective unitary group PU(ℋ))
The projective unitary group PU(ℋ) on an infinite-dimensional separable Hilbert space is:
a Banach Lie group in its norm topology, and as such well-pointed by Ex.;
no longer a Banach space in its weak/strong operator topology, but nevertheless still well-pointed in this case, by this Prop..
Textbook accounts:
Glen Bredon, Section VII.1 of: Topology and Geometry, Graduate texts in mathematics 139, Springer 1993 (doi:10.1007/978-1-4757-6848-0, pdf)
Tammo tom Dieck, Algebraic topology. European Mathematical Society, Zürich (2008) (doi:10.4171/048)
Last revised on May 6, 2022 at 12:49:22. See the history of this page for a list of all contributions to it.