## Category → math

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):

(ported from wiki) Certain notable problems that I don’t want to look through a zillion pages to find. Iran TST 1996, notoriously reposted at least 35 times on AoPS (okay, many of these are actually modifications): If $$x, y, z > 0$$ then $(xy+yz+zx)\left(\frac{1}{(x+y)^2} + \frac{1}{(y+z)^2} + \frac{1}{(z+x)^2}\right) \geq \frac{9}{4}$ ISL 1988 #4: if $$1, 2, \ldots, n^2$$ are placed in a $$n \times n$$ chessboard, some two adjacent numbers differ by at least $$n$$ USAMO 1995.

### 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

$f_\epsilon(x) = \begin{cases} 0 & \text{if } |x| \geq \epsilon \\ 1 - \left|\frac{x}{\epsilon}\right| &\text{if } |x| < \epsilon \end{cases}.$