Teaching the Baire Category Theorem in an Intro Point-set Course: a mini-rant.

January 5, 2012

Whenever I talk to students who have had basic point-set topology, I’m not surprised if they haven’t seen the BCT; it’s near the end of the point-set part of Munkres, and one natural stopping point for a semester-long point-set class is Urysohn’s lemma which is done before BCT. I don’t remember learning the BCT in point-set when I took it and it wasn’t until Functional Analysis that I had to study it again. This is a shame for a few reasons, but I wanted to point out just two.

1. This theorem is able to be proved (or at least stated) after complete metric spaces are defined and would take only around one lecture to do with applications. Depending on the speed of the professor and the way one works through the book, one probably gets to metric spaces around the middle of the semester. The students have had analysis, and are, therefore, familiar with completeness (or, at worst, they’re comfortable with Cauchy sequences!) so it is not difficult to define complete metric spaces. What else would we need? Nowhere-dense sets. That’s all.
2. This can be a great “visual” theorem in some spaces. One common problem point-set topology students seem to have is that they will work through definitions without developing any kind of intuition behind what something is (eg, limit points). Therefore, it’s nice to learn properties of spaces we know and love. The BCT is easily applied to the reals, the irrationals, the rationals, and the cantor set. It also is a great brain-stretching exercise to think about how “close” points can be in a nowhere dense set on the real line.

This may be a non-issue for those of you whose point-set class included this, but I wanted to point is out nonetheless. I would go as far as saying that this is “more important” to cover in a first course in point-set than the proof of Urysohn’s lemma [which is the first “deep” result in Munkres]. Of course, the proof is important, but in an introductory course I feel that proving Urysohn’s lemma for metric spaces and stating it for general spaces is a better way to go. Feel free to disagree and yell at me!

Edit: This rant used to be part of the BCT post I made recently, but because it’s not really mathematics and more about teaching I decided to post it on its own so I don’t have to subject everyone to reading my rants.