### Gauge Spaces and the Stone-Cech Compactification

#### Posted by Mike Shulman

If you’d like a change of pace from category theory, physics, and philosophy, have a look at these notes that I just posted to the arXiv:

Probably the best way to introduce them is to quote from the second introduction (the one addressed to people who know what a topological space is):

The goal of these notes is to develop the basic theory of the Stone-Cech compactification without reference to open sets, closed sets, filters, or nets. In particular, this means we cannot use any of the usual definitions of topological space. This may seem like proposing to run a marathon while hopping on one foot, but I hope to convince you that it is easier than it may appear, and not devoid of interest.

In brief, the approach is:

Use gauge spaces — spaces equipped with a family of pseudometrics — as our notion of “space”. I think these are almost as easy to conceptualize as metric spaces.

Define “compact” to mean “totally bounded and complete”.

Define “complete” using a gauge-space version of Lawvere’s completeness criterion for metric spaces that comes from viewing them as enriched categories.

These notes are based on a course I gave at Mathcamp this summer (and also four summers ago). I wanted to talk about the Stone-Cech compactification, but I didn’t want to require the students to already know what a topological space was — hence the decision to use gauge spaces. I think it mostly worked out quite nicely! (Of course, we didn’t get to nearly everything in here during the Mathcamp classes; plus I’ve added a good deal more in polishing up the notes to post.) There are plenty of exercises, some of which introduce other interesting concepts like proximity, or classical definitions like “open set” and “topological space”. Enjoy!

## Re: Gauge Spaces and the Stone-Cech Compactification

All the category theoretic framing in the note made me wonder, do gauge spaces admit a description as enriched categories, analogous to Lawvere quasimetric spaces?

Also, did I correctly understand that the axiom of choice was not used for the proof of compactness of $X^{C(X,I)}$, but rather only for the extension of continuous maps from $X$ to $\beta X$? Is this something that happens for ordinary metric spaces as well? Tychonoff’s theorem is constructively valid? What is the significance of that?