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quasi-topological space

Quasi-topological spaces

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

Algebraic topology

Quasi-topological spaces

Idea

The category of quasi-topological spaces was proposed in Spanier 1963 as a substitute for the category Top of ordinary topological spaces, in order to serve as a convenient category for the purposes of algebraic topology. In particular, quasi-topological spaces form a complete and cocomplete cartesian closed category.

Today, quasi-topological spaces seem to be regarded mostly as a historical curiosity, perhaps because working topologists were never comfortable with the set-theoretic issues that accompany them. In retrospect, however, they are an impressive testament to the conceptual insight of Spanier into ideas of topos theory which were at the time (early 1960’s) barely in the air, and even not quite born yet (being an early example of quasitopos, whose name perhaps derives from Spanier’s notion, compare Dubuc & Español 2006, p. 12).

Definition

Let 𝒞ℋ\mathcal{C H} be the category of compact Hausdorff spaces. This may be regarded as a (large) site when equipped with the Grothendieck topology of finite open covers, in fact a concrete site.

Definition

A quasi-topological space is a (small-set valued) concrete sheaf on 𝒞ℋ\mathcal{C H}.

The (super-large) category of quasi-topological spaces is a quasitopos (although this is not immediately obvious for size reasons — in particular, it is probably not a Grothendieck quasitopos). In particular, it is a locally cartesian closed category.

References

Last revised on August 24, 2021 at 09:30:39. See the history of this page for a list of all contributions to it.