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):

- 2.33,
**space-independence**: still compact regardless of what space you consider it part of - 2.34,
**closed** - 2.35,
**hereditary**: closed subsets are compact - 2.36, corollary
**telescoping descent**: a descending sequence of nonempty compact \(K_1 \supseteq K_2 \supseteq K_3 \supseteq \cdots\) contains at least one point- 2.36 actually says if the K
_{i}are compact and any finite subcollection has nonempty intersection, then the entire colleciton has nonempty intersection. - 3.10(b),
**shrinking convergence**: if the \(K_i\) get infinitely small, they contain exactly one point.

- 2.36 actually says if the K
- 2.37,
**infinite accumulation**: infinite subsets have at least one limit point- 3.6,
**sequential compactness**: infinite sequences have convergent subsequences- (Note from the future: This is the actual name of this property in topology. In general, neither compactness nor sequential compactness imply the other, but they are equivalent in metric spaces.)

- 3.6,

### Chapter 4: Continuity

- 4.8: A function is continuous iff it inverts open sets to open sets.
- Corollary: A function is continuous iff it inverts closed sets to closed sets

- 4.9: A continuous function maps compact sets to compact sets.
- 4.19: A continuous function on a compact domain is uniformly continuous.

### Chapter 5: Differentiation

- 5.6(b) is the differentiable function with discontinuous derivative.
- 5.12: derivatives attain intermediate values.
- 5.15,
**Taylor’s Theorem**(not exact form): Let \(f\) be \(n\) times differentiable near \(\alpha\). Then \(f\) is equal to the degree-\(n\) Taylor polynomial near \(\alpha\) plus a remainder \(h(x)(x - \alpha)^n\) with \(h\) tending to 0 at \(\alpha\).- Rudin states the theorem with the
**Lagrange form**of the remainder: at \(x\) near \(\alpha\), \(f\) is equal to the degree-\(n\) Taylor polynomial*after*the coefficient of the degree-\(n\) term is fudged into using \(f^{(n)}(\xi_L)\) (instead of \(f^{(n)}(x)\) for some \(\xi_L\) between α and x. - Also the theorem’s conditions are tricky about what exactly is required at endpoints of intervals.

- Rudin states the theorem with the

### Chapter 8: Some Special Functions

8.8,

\(f\) has an infimum of magnitude and attains it. Recenter it at that infimum and note that it behaves like its smallest-degree nonconstant monomial nearby, which means we can perturb its value towards 0 if the infimum is nonzero. So the infimum is zero.**algeraic completeness of \(\mathbb{C}\)**.- 8.14: A “Lipschitz-continuous-at-a-point” function is approached by its Fourier series.
- 8.15: A \(2\pi\)-periodic continuous function is uniformly approximate-able by trigonometric polynomials.
- 8.16,
**Parseval’s theorem**:- the integral of the square of a function’s absolute difference from its Fourier series tends to 0
- the “dot product” of two functions tends to the dot product of their Fourier series coefficients (with fudge factors due to non-orthonormality)

8.19: the gamma function is the only extension of the factorial function with a convex log.