Proving that sequentially compact spaces are compact.

I remember seeing this proof somewhere (perhaps here, but I don't remember where) that goes something like this.

Suppose $X$ is sequentially compact, and by contradiction suppose $\{U_n\}$ is a countable open cover with no finite subcover. Then for any positive integer $n$, the set $\{U_i : i \le n\}$ is not an open cover, so there exists $x_n \notin \bigcup_{i \le n} U_i$. Hence, we obtain sequence, and by sequential compactness, there exists a subsequence $x_{n_j}$ that converges to $a \in X$. However, $a \in U_k$ for some positive integer $k$ and by construction, $x_{n_j} \notin U_k$ if $n_j \ge k$. This is a contradiction.

Doesn't this only prove every countable open cover must have a finite subcover?

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In general, sequential compactness neither implies, nor is implied by, compactness; but for metric spaces, they are equivalent (Wikipedia reference). –  Zev Chonoles Jun 29 '12 at 6:34
Now that I think of it, the result was probably "Sequential compactness implies countable compactness" - not this. –  Dom Jun 29 '12 at 6:41
Observe that for metric spaces the notions are indeed equivalent, since we can prove that sequentially compact (and compact) metric spaces are second-countable, so every open cover can be replaced with basic cover, but we have a countable basis so we have the open cover is countable - and the proof you gave follows. –  Asaf Karagila Jun 29 '12 at 7:02
See form [8 AE] at consequences.emich.edu/conseq.htm. $\:$ –  Ricky Demer Jun 29 '12 at 7:32

Maybe you are unclear for the definitions of the compactness and countably compactness.

1. compactness = for any open cover, there is a finite open subcover which covers the whole space.
2. countably compactness = for any countable open cover, there is a finite open subcover which covers the whole space.

By your proof we only see that every sequentially compact space is countably compact, which you can see the Theorem 3.10.30 of the Engelking's book:) However not every sequentially compact space is compact.

For example, the topological space $\omega_1$ with the order topology is a sequentially compact space but not a compact space. In the first countable space (in fact, it's only need sequential space), sequentially compace = countably compact. As we know, the space is a first countable space and countably compact, therefore, it is a sequentially compact. But, it is not a compact space:)

By your method ( we use it much as a topologist ), the sequence your obtained is a closed discrete subspace. It hasn't a cluster point:) So it's a contradiction with sequential compactness.

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