Note: My 2009 self wrote this. It is preserved for
historical interest and amusement, and as a testament to how far I’d go
to typeset math equations in ASCII art.
I have just come back from a week of continuous mathematics, board
gaming, and hitting people with sticks to ensure that they get up in
time. Yippee! Those problems sure do get around.
Sample problem: This problem rocks!
Brevity has been chosen over accuracy because the whole point is that
you should know this stuff already.
Chapter 2: Basic Topology
(+ some Ch. 3)
An isolated point of E is in E but not a limit point
of it. E is perfect if it is exactly equal to its set
of limit points. Equivalently, it is closed and has no isolated points.
Ex. 2.44: The Cantor set is perfect.
A compact set is a set for which every open cover
has a finite subcover.
Compactness or compact sets have these properties (with made-up
names):
Limit + Limit
Cheaply, using an Iverson
bracket expression:
\[
\begin{aligned}
\lim_{a \to \infty} \lim_{b \to \infty} [a > b] &= 0 \\
\lim_{b \to \infty} \lim_{a \to \infty} [a > b] &= 1
\end{aligned}
\]
For more continuity, use \(\frac{a}{a +
b}\) instead (Rudin Example 7.2).
Uniform convergence on one limit suffices to allow this exchange,
almost by definition.
Limit + Continuity
Let
\[
f_\epsilon(x) = \begin{cases} 0 & \text{if } |x| \geq \epsilon \\ 1
- \left|\frac{x}{\epsilon}\right| &\text{if } |x| < \epsilon
\end{cases}.
\]