A Story about the Past Three Posts

I consider the previous three posts I wrote as a trilogy of sorts, and it was my procrastination of posting the last one that motivated me to start this blogging project. I’ve been meaning to write up yesterday’s post since June.During a conference at Hebei Normal University in Shijiazhuang, I gave a talk where George Elliott questioned, among other things, the fact that integer-valued continuous functions form a free abelian group. I gave the non-constructive answer that I gave at the beginning of the yesterday’s post. Later, during the conference banquet, I spoke with him about this problem, where he proposed a more constructive solution for the Cantor set. It was a bit difficult to understand without any visual aids, and since I was not entirely sober at the time. After this banquet, I spoke to my friend and colleague Wei Sun about this problem and he said that he had considered a similar solution for the Cantor set.

During a conference at Hebei Normal University in Shijiazhuang, I gave a talk where George Elliott questioned, among other things, the fact that integer-valued continuous functions form a free abelian group. I gave the non-constructive answer that I gave at the beginning of the yesterday’s post. Later, during the conference banquet, I spoke with him about this problem, where he proposed a more constructive solution for the Cantor set. It was a bit difficult to understand without any visual aids, and since I was not entirely sober at the time. In retrospect, I suspect he was correct. After this banquet, I spoke to my friend and colleague Wei Sun about this problem and he said that he had considered a similar solution for the Cantor set. It was much clearer with visual aids.

At this point, I thought this proof would make a nice little blog post once I was able to generalize the result to all compact Hausdorff spaces. But the generalization wasn’t immediately clear since both people talked about the Cantor sets in terms of the standard middle-thirds construction. This reminded me of the proof in Davidson that I talked about two days ago, and I switched views from the middle-thirds construction to a picture of the infinite complete binary tree (given by the Bratteli diagram) for the Cantor set. With the picture of rooted trees, it became clear how to do this for totally disconnected spaces and by taking the appropriate quotient, how to do it for all (compact Hausdorff) spaces.

Being reminded of that Davidson proof led me down a rabbit hole of how I reconciled that proof with this picture, which brought me to the Stone-Weierstrass theorem: the post from three days ago. So the ideas came in reverse from how I posted them, which seems how a lot of math works in my experience.

I hope you’ve enjoyed that trilogy of posts. They were ideas that I enjoyed thinking about and enjoyed writing about.

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About minimalrho
Unemployed guy with a PhD in math.

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