nLab
well-pointed topological space

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Homotopy theory

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

Contents

Definition

A pointed topological space (X,x)(X,x) is called well-pointed if the base-point inclusion {x}X\{x\} \xhookrightarrow{\;} X is a Hurewicz cofibration (e.g. Bredon 1993, VII, Def. 1.8).

A topological group is called well-pointed if it is so at its neutral element, hence if {e}G\{\mathrm{e}\} \xhookrightarrow{\;} G is a 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.

Examples

Example

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.

(Immediate by the discussion of examples of Hurewicz cofibrations).

Example

(the projective unitary group PU(ℋ))
The projective unitary group PU(ℋ) on an infinite-dimensional separable Hilbert space is:

References

Textbook accounts:

Last revised on September 19, 2021 at 12:12:17. See the history of this page for a list of all contributions to it.