# 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)