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September 14, 2012

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!

Posted at September 14, 2012 1:02 AM UTC

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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)X^{C(X,I)}, but rather only for the extension of continuous maps from XX to βX\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?

Posted by: Ziggurism on May 20, 2018 5:44 PM | Permalink | Reply to this

Re: Gauge Spaces and the Stone-Cech Compactification

I addressed your first question in the notes (p4):

Clementino, Hofmann, and Tholen [1,3] have shown that uniform spaces, and their completions [2], can be described in a framework which generalizes Lawvere’s. I have not been able to find such a framework which reproduces gauge spaces exactly, although they are a special case of the prometric spaces of [3].

As for your second question: I didn’t have my constructivist hat on when I was writing these notes, but I believe that you’re right that I didn’t use AC to prove compactness of the Stone-Cech compactification. In fact, I think the extension of continous maps from XX to βX\beta X even works without AC as long as the target space YY is compact Hausdorff, since in that case the “choices” are unique. There does seem to be a fundamental use of AC in the proof in section 17 that βX\beta X surjects onto all other compactifications, though. (And the law of excluded middle is everywhere — it could probably be removed in some cases, but note that without it the real numbers don’t have all suprema and infima.)

I’m not sure what you mean by “Is this something that happens for ordinary metric spaces as well?” since metric spaces are a special case of gauge spaces, and the construction doesn’t stay in the world of metric spaces; so it seems to me the answer is either trivially yes or trivially no, depending on what the question is. I don’t immediately see how this fact implies any general form of Tychonoff’s theorem. Moreover, note that the definition of “compact” here, namely “complete and totally bounded”, is not constructively equivalent to the usual definition! Errett Bishop also chose to use “complete and totally bounded” as his definition of “compact” (he only considered metric spaces) for this very reason that it is actually better-behaved constructively.

Posted by: Mike Shulman on May 21, 2018 9:30 PM | Permalink | Reply to this

Re: Gauge Spaces and the Stone-Cech Compactification

I addressed your first question in the notes (p4):

Ah yes I see. Sorry, I might’ve skipped the introduction and gone right to the definitions.

Clementino, Hofmann, and Tholen [1,3] have shown that uniform spaces, and their completions [2], can be described in a framework which generalizes Lawvere’s.

Ok so having looked at Clementino, Hofmann, and Tholen [3], the idea is that since enriched categories are monoids in the category of matrices valued in the enrichment category, then whereas Lawvere metric spaces are monoids in matrices valued in R, pro-metric spaces are monoids of the pro-category of same.

For me that’s not as accessible and intuitive as Lawvere’s revelation about metric spaces. But that cubical diagram with preorders, topological spaces, approach spaces, and metric spaces, with arrows to pro-preorders, pro-topological spaces, pro-approach spaces, and pro-metric spaces seems very cool, so maybe it’s worth trying to penetrate.

I’m not sure what you mean by “Is this something that happens for ordinary metric spaces as well?” since metric spaces are a special case of gauge spaces, and the construction doesn’t stay in the world of metric spaces; so it seems to me the answer is either trivially yes or trivially no, depending on what the question is.

Right. The answer is trivially yes.

It’s just that the equivalence of the axiom of choice and Tychonoff’s theorem is so well-known for topological spaces. And then there’s the lesser known fact that Tychonoff’s theorem for locales is valid in ZF. Since metric spaces are so much more familiar, something in the same family as topological spaces, I was surprised that I had never heard about the status of Tychonoff for metric spaces. If I had known that Tychonoff for metric spaces did not require AC, maybe I wouldn’t have been so impressed when I learned of locales…

Posted by: Ziggurism on May 27, 2018 8:03 PM | Permalink | Reply to this

Re: Gauge Spaces and the Stone-Cech Compactification

For me that’s not as accessible and intuitive as Lawvere’s revelation about metric spaces.

To be sure. But more complicated things are always more complicated. (-:

And then there’s the lesser known fact that Tychonoff’s theorem for locales is valid in ZF.

And even without excluded middle!

If I had known that Tychonoff for metric spaces did not require AC

Even if we grant that defining “compact” to mean “complete and totally bounded” for (pro)metric spaces is analogous to the relevant definition of compactness for locales (which I’m dubious of — compactness for locales is much more obviously a translation of the usual notion of compactness for topological spaces), how are you proposing to deduce “Tychonoff for metric spaces” from compactness of their Stone-Cech compactifications?

Posted by: Mike Shulman on May 27, 2018 8:25 PM | Permalink | Reply to this

Re: Gauge Spaces and the Stone-Cech Compactification

Even if we grant that defining “compact” to mean “complete and totally bounded” for (pro)metric spaces is analogous to the relevant definition of compactness for locales (which I’m dubious of — compactness for locales is much more obviously a translation of the usual notion of compactness for topological spaces),

I think I hear what you are saying. We should not get confused and mix up Tychonoff’s theorem with the Heine-Borel theorem. Heine-Borel says every cover has finite subcover iff complete and totally bounded. It does not require choice. And maybe that’s all we saw in your gauge space exposition. Somehow showing completeness and total boundedness of a product of complete and totally bounded spaces does not require the same choice invocations that showing the product of subcover compact spaces is subcover compact does.

But I wonder why?

Posted by: Ziggurism on May 31, 2018 9:21 PM | Permalink | Reply to this

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