Contents

# Contents

## 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. )

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

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:

Review:

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

Last revised on January 8, 2021 at 07:52:06. See the history of this page for a list of all contributions to it.