Point topology 定理

Bolzano-Weistrass theorem

• If a bounded set $S \subseteq \mathbb{R}^n$ contains infinity many points, then there is at least one point in $\mathbb{R}^n$ which is a limit point of $S$.

• Every sequence in $S$ has a subsequence that converges a point of $S$.

Cantor intersection theorem

• $\lbrace Q_1, Q_2, \cdots,\rbrace$ be a countable collection of nonempty sets in metric space $(X, d)$ such that

  • $Q_{k+1} \subseteq Q_k$, $k=1,2,\cdots,$.

  • Each set $Q_k$ is bounded and $Q_1$ is closed set.

  • $\Rightarrow$ $\cap_{k=1}^{\infty}Q_k$ is closed and nonempty set.

Lindelof covering theorem

• $(X,d)$ is metric space, and $S \subseteq X$.

  • $G$ is an open covering of $S$.

  • then there is a countable subcollection of $G$ which also covers $S$.

• 此定理證明了在metric space $X$中的任意集合$S$如果為open covering (countable or uncountable)，則$S$必可被可數的集合open covering。

Heine-Borel theorem

• $E \subseteq \mathbb{R}^n$, then the following properties are equivalent:

  • $E$ is closed and bounded set.

  • $E$ is compact set.

  • Every infinite subset of $E$ has a limit point in $E$.