nLab one-point compactification intertwines Cartesian product with smash product

Statement

Proposition

(one-point compactification of product space is smash product of the compactified factors)

On the subcategory $Top_{LCHaus}$ in Top of locally compact Hausdorff spaces with proper maps between them, the functor of one-point compactification (Prop. )

$$(-)^{cpt} \;\colon\; Top_{LCHaus} \longrightarrow Top^{\ast/}$$

1. sends coproducts, hence disjoint union topological spaces, to wedge sums of pointed topological spaces;

2. sends Cartesian products, hence product topological spaces, to smash products of pointed topological spaces;

hence constitutes a strong monoidal functor for both monoidal structures of these distributive monoidal categories in that there are natural homeomorphisms

$$\big( X \sqcup Y \big)^{cpt} \;\simeq\; X^{cpt} \vee Y^{cpt} \,,$$

and

$$\big( X \times Y \big)^{cpt} \;\simeq\; X^{cpt} \wedge Y^{cpt} \,.$$

This is briefly mentioned in Bredon 93, p. 199. The argument is spelled out in: MO:a/1645794, Cutler 20, Prop. 1.6.

References

Basic accounts:

• Glen Bredon, p. 199 of: Topology and Geometry, Graduate Texts in Mathematics 139, Springer 1993 (doi:10.1007/978-1-4757-6848-0, pdf)

Review:

• Tyrone Cutler, The category of pointed topological spaces (2020) [pdf, pdf]