nLab well-pointed topological space

Definition

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.

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:

• 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..

References

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)