The Bend in the River

Read-along: Rudin's PoMA: Theorems from chapter 2

Visit the main page for this week

Here is a list of theorems, sans proofs, of the theorems in the latter part of chapter 2.

Definitions. metric space, neighbourhood (ball), limit point, isolated point, closed, interior point, open, complement, perfect, bounded, dense.

2.19. Every neighbourhood is an open set.

2.20. If $ p $ is a limit point of $ E $, then each neighbourhood of $ p $ contains infinitely many points of $ E $.

2.20 (corollary). A finite set has no limit points.

2.22. Let $ \{E_\alpha\} $ be a family of sets. Then $\left(\cup_\alpha E_\alpha\right)^\complement = \cap_\alpha E_\alpha^\complement$.

2.23. $ E $ is open iff $ E^\complement $ is closed.

2.24. The collection of open sets of a metric space $ X $ forms a topology:

1. $ X $ is open.
2. $ \emptyset $ is open.
3. $ X $ is closed.
4. $ \emptyset $ is closed.
5. If $ I $ is an arbitrary index set and $ \{E_\alpha\}_{\alpha \in I} $ is a family of open sets, then $ \cup_\alpha E_\alpha $ is open.
6. If $ I $ is an arbitrary index set and $ \{F_\alpha\}_{\alpha \in I} $ is a family of closed sets, then $ \cap_\alpha F_\alpha $ is closed.
7. If $ N \in \mathbb{N} $ and $ \{E_i\}_{i = 1}^N $ is a family of open sets, then $ \cap_\alpha E_\alpha $ is open.
8. If $ N \in \mathbb{N} $ and $ \{F_i\}_{i = 1}^N $ is a family of closed sets, then $ \cup_\alpha F_\alpha $ is closed.

2.25. There exists an infinite (countable, even) family $ \{E_i\}_{i = 1}^N $ of open sets in $ \mathbb{R} $ such that $ \cap_\alpha E_\alpha $ is not open.

2.27. Let $ X $ be a metric space and $ E \subseteq X $. Then:-

1. $ \overline{E} $ is closed.
2. $ E = \overline{E} $ iff $ E $ is closed.
3. $ \overline{E} \subseteq F $ for every closed set $ F \subseteq X $ such that $ E \subseteq F $.
4. (Hence) $ \overline{E} $ is the smallest (wrt set inclusion) closed set in $ X $ containing $ E $.

2.28. Let $ E $ be a non-empty set of real numbers bounded above. Let $ y = \sup E $. Then $ y \in \overline{E} $. Hence $ y \in E $ if $ E $ is closed.

Question. Is the converse true - if $ \sup E \in E $, does it follow that $ E $ is closed?

2.30. Suppose $ Y \subseteq X $. A subset $ E $ of $ Y $ is open relative to $ Y $ iff $ E = Y \cap G $ for some open $ G \subseteq X $.

Definition. compactness.

2.33. Suppose $ K \subseteq Y \subseteq X $. Then $ K $ is compact relative to $ X $ iff $ K $ is compact relative to $ Y $.

2.34. Compact subsets of a metric space are closed.

2.35. Closed subsets of compact sets are compact.

2.35 (corollary). If $ F $ is closed and $ K $ is compact then $ F \cap K $ is compact.

2.36. If $ \{K_\alpha\} $ is a family of compact subsets of a metric space $ X $ such that every finite subfamily has nonempty intersection, then $ \cap_\alpha K_\alpha $ is nonempty.

Counterexample. Show that if ‘compact’ is replaced with ‘closed’ then the above theorem is false.

2.36 (corollary). If $ \{K_n\} $ is a sequence of non-empty compact sets such that $ K_n \supseteq K_{n + 1} $ for each $ n \in \mathbb{N} $, then $ \cap_1^\infty K_n $ is non-empty.

Finite intersection property. Let $ X $ be a metric space. $ X $ is compact iff for every collection of closed sets $ \{F_\alpha\} \subseteq \mathcal{P}(X) $,

2.37. If $ E $ is an infinite subset of a compact set $ K $, then $ E $ has a limit point in $ K $.

2.38. If $ \{I_n\} $ is a sequence of intervals in $ \mathbb{R} $ such that $ I_n \supseteq I_{n + 1} $ for each $ n \in \mathbb{N} $, then $ \cap_1^\infty I_n $ is non-empty.

2.39. Let $ k \in \mathbb{N} $. If $ \{I_n\} $ is a sequence of $ k$-cells such that $ I_n \supseteq I_{n + 1} $ for each $ n \in \mathbb{N} $, then $ \cap_1^\infty I_n $ is non-empty.

2.40. Every $ k$-cell is compact.

2.41 (Heine-Borel). Let $ E \subseteq \mathbb{R}^k $. TFAE:

1. $ E $ is closed and bounded.
2. $ E $ is compact.
3. Every infinite subset of $ E $ has a limit point in $ E $.

2.42 (Weierstraß). Every bounded infinite subset of $ \mathbb{R}^k $ has a limit point in $ \mathbb{R}^k $.

2.43. Let $ P \subseteq \mathbb{R}^k $ be non-empty and perfect. Then $ P $ is uncountable.

2.43 (corollary). Every closed interval is uncountable.

2.44. The Cantor set is perfect.

Definition. connected.

2.47. A subset $ E \subseteq \mathbb{R} $ is connected iff the following holds: