nLab
one-point compactification intertwines Cartesian product with smash product

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

Contents

Statement

Proposition

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

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

() cpt:Top LCHausTop */ (-)^{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

(XY) cptX cptY cpt, \big( X \sqcup Y \big)^{cpt} \;\simeq\; X^{cpt} \vee Y^{cpt} \,,

and

(X×Y) cptX cptY cpt. \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.