Section 27: Compact Subspaces of the Real Line

1. Generalized Extreme Value Theorem. If $f:X\to Y$ is a continuous function from a compact space to an ordered set in the order topology, then there are $x_1$ and $x_2$ : $f(x_1)\le f(x)\le f(x_2)$ for all $x\in X$ .
2. Ordered sets and compactness:
3. A compact ordered set has the least and the largest elements.
4. An ordered set satisfies the least upper bound property iff every closed interval is compact.
5. An ordered set is compact iff it has the least and the largest elements and satisfies the least upper bound property.
6. Consider $[0,1)\cup[2,3]$ : it is compact in the order topology, but not compact in the subspace topology.
7. A subspace of $\mathbb{R}^n$ is compact iff it is closed and bounded in the standard or square metric.
8. Uniform continuity: let $(X,d)$ and $(Y,d')$ be metric spaces, then
9. The distance from a point $x$ to a set $A$ is $\inf\{d(x,a)|a\in A\}$ .
10. $|d(x,A)-d(y,A)|\le d(x,y)$
11. The Lebesgue Number Theorem: If $(X,d)$ is compact and $\mathcal{A}$ is a covering of $X$ then there is $\delta$ (the Lebesgue number) such that if the diameter of a set is less than $\delta$ then it is contained within a set in $\mathcal{A}$ .
12. $f:X\to Y$ is uniformly continuous if for every $\epsilon>0$ there is $\delta>0$ such that $d(x,x')<\delta$ implies $d'(f(x),f(x'))<\epsilon$ .
13. Uniform continuity theorem: A continuous function from a compact metric space to a metric space is uniformly continuous.
14. A nonempty compact Hausdorff space without isolated points is uncountable.
15. For a countable sequence of points find a sequence of nested closed sets such that their intersection does not contain any points of the sequence.
16. If $\{A_n\}$ is a countable collection of closed subsets of a compact Hausdorff space and each set has empty interior, then their union has empty interior as well.
17. A connected metric space containing at least two points is uncountable.
18. The Cantor set: $[0,1]-\cup_{n\ge 1}\cup_{k\ge 0}\left(\frac{1+3k}{3^n},\frac{2+3k}{3^n}\right)$ .
19. Compact, totally disconnected without isolated points and uncountable.