Notes on a Few Abnormalities in Analysis
I have my last exam for my Analysis class tomorrow. What surprised me most about the class was the amount of counterintuitive examples that arise in every area. Here's a few I like.
The Heine-Borel theorem says that every closed and bounded subset of the reals is compact. However this does not extended to other metric spaces. Take X = (0,1). S = (0,1/2] is a closed and bounded subset of X, but it is not compact in X. Dang.
The Cantor Set. Everything about it is counterintuitive. Cantor was really trying to make life harder for people. It's closed, contains no positive length interval, and uncountable. The application of his diagonal argument to show the set is uncountable is great.
Compact spaces are nice. Make life a lot easier. Things fall apart when the space is not compact. Your sequences may converge but not to an element in your space.
A set can be S can be the set of discontinuities for a function f iff it is the union of countably many closed subsets. At first I got this but then I realized it means there can be some function that is continuous at all the irrationals and discontinuous at all the rationals. And then I got home and looked it up: https://en.wikipedia.org/wiki/Thomae%27s_function .
I learned about conditional and absolute convergence of series in high school. I was unaware though that if a series is only conditionally convergent then you can find a rearrangement which diverges along with different arrangements that sum to every real number. It almost feels like cheating, saying that a given series converges when in reality you can shuffle it around and it's an absolute mess.
Analysis has been fun. I've enjoyed thinking a lot harder about things I learned a few years ago and took for granted. We can always understand things on a deeper level.