nLab
point-free topology

Context

Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Basic homotopy theory

Theorems

Contents

Idea

Point-free topology (often referred to as pointless topology Johnstone 83 ) is any formulation of topology not based on the notion of topological space as a set of points equipped with extra structure. (A pointless space must still be this set with extra stuff, of course, as long as there is a functor mapping a space to its set of points.) Pointless topology has points, but they are not fundamental; and a particular space may well have no points and yet be far from empty.

In locale theory, for example, one studies the set of open subspaces (with the extra structure of a frame) as the fundamental notion. In formal topology, one studies a set of basic open subspaces (with the extra structure of a posite with positivity, although the isomorphisms of formal spaces don't respect these sets).

In contrast, the traditional way of doing topology using points may be called pointwise topology.

Definition

In the interest of considering whether a formulation of topology is pointless or not, I offer the following sociomathematical suggestion at a definition:

Working in a given logical context \mathcal{L}, suppose that one defines a category (or (,1)(\infty,1)-category) SS, whose objects one thinks of as spaces and whose morphisms one thinks of as continuous maps. Suppose further that one intends Hom(X,Y)Hom(X,Y) to be the set (or \infty-groupoid) of all continuous maps from XX to YY (whereas Ob(S)Ob(S) need not be the class of all spaces). Also suppose that SS has a terminal object ptpt, which one interprets as the point.

Definition

The above is a pointwise formulation of topology if it is provable (in \mathcal{L}) that ptpt is a generator in SS but pointless if this is not provable. (One could have stronger notions of pointlessness by asking that this be refutable; if using intuitionistic logic, this could be further strengthened.)

References

An introduction to locale theory is

  • Peter Johnstone (1983); The point of pointless topology; Bull. Amer. Math. Soc. (N.S.) Volume 8, Number 1, 41–53; Euclid.

This is, in its own words, to be read as the trailer for Johnstone’s book Stone Spaces, which see.

For formal topology, see

  • Giovanni Sambin (2001); Some points in formal topology; pdf.

  • Erik Palmgren, From Intuitionistic to Point-Free Topology: On the Foundation of Homotopy Theory, Logicism, Intuitionism, and Formalism Volume 341 of the series Synthese Library pp 237-253, 2005 (pdf)

Revised on May 21, 2017 06:30:50 by Urs Schreiber (92.218.150.85)