nLab pointless topology

Context

Topology

topology

algebraic topology

Contents

Idea

Pointless topology 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 $(\infty,1)$-category) $S$, whose objects one thinks of as spaces and whose morphisms one thinks of as continuous maps. Suppose further that one intends $Hom(X,Y)$ to be the set (or $\infty$-groupoid) of all continuous maps from $X$ to $Y$ (whereas $Ob(S)$ need not be the class of all spaces). Also suppose that $S$ has a terminal object $pt$, which one interprets as the point.

Definition

The above is a pointwise formulation of topology if it is provable (in $\mathcal{L}$) that $pt$ is a generator in $S$ 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

Revised on May 7, 2012 19:55:07 by Toby Bartels (64.89.53.248)