Pardon the alliteration. I spent the second half of last week preparing for my talk on Friday. I gave a two-hour introduction to General Topology. I don't think it went that badly, except for a few moments where I blew a proof (actually, I just left out a quantifier that the proof hinged on. Kelley did the same, but I clearly wasn't quite thinking straight when I copied the proof out to present), and a couple definitions.
Speaking of definitions, it's interesting that one can give an engaging talk for quite a long time only covering basic definitions without providing any real results. This was the second talk I've given for PL Jr; and the second for which I've done precisely that. The last one was on Universal Algebra. I've heard it said before that there are 3 kinds of mathematicians: Algebraists, Topologists, and Analysts. I'll be very surprised if I hit the trifecta.
I think the best line of the day was when someone was trying to claim that the intersection of two open intervals could contain a single point, and hence be closed. I responded by pointing out that no matter how small the intersection was, it was either empty or contained uncountably many points. "And," I contended, "I contend that a set with uncountably many points is not a singleton."
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