Point topology 定理
Bolzano-Weistrass theorem
If a bounded set contains infinity many points, then there is at least one point in which is a limit point of .
Every sequence in has a subsequence that converges a point of .
Cantor intersection theorem
be a countable collection of nonempty sets in metric space such that
, .
Each set is bounded and is closed set.
is closed and nonempty set.
Lindelof covering theorem
is metric space, and .
is an open covering of .
then there is a countable subcollection of which also covers .
此定理證明了在metric space 中的任意集合如果為open covering (countable or uncountable),則必可被可數的集合open covering。
Heine-Borel theorem
, then the following properties are equivalent:
is closed and bounded set.
is compact set.
Every infinite subset of has a limit point in .