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Thoughts in Low Places

December 12, 2022

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. 



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