On 9/29/16, we discussed three notions of compactness in equipped with the usual topology:
(1) closed and bounded
(2) subsequential compactness
(3) every open cover admits a finite subcover
We will tacitly assume that this is the topology we're giving for the remainder of this post.
- We proved that (1) and (2) are equivalent.
- Statements (2) and (3) are equivalent in general metric spaces.
- (1) is not necessarily equivalent to (2) or (3) in other settings (and even non-contrived settings: that is, settings which are not just around for the sake of counterexample. There is an abundance of examples arising from basic objects of study in functional analysis.).