1617-257/TUT-R-3

On 9/29/16, we discussed three notions of compactness in $\mathbb {R} ^{n}$ 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 $\mathbb {R} ^{n}$ for the remainder of this post.

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- 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 non-contrived settings (i.e. Settings which are not just produced for the sake of counterexample. There is an abundance examples arising from basic objects of study in functional analysis.).