Their definitions seem to be very similar, what is the difference?
Here are the definitions I'm using:
Heine-Borel: Every open cover for $E$ admits a finite subcover.
Countable compactness: Let $(G_i)^{\infty}_1$ be a sequence of open sets in metric space $M$. Then $E \subseteq M$ is countably compact whenever $E \subseteq \bigcup_{i=1}^{\infty}G_i$ there exists finite $n_1…n_k$ such that $E \subseteq \bigcup^k_{i=1}G_{n_k}$.
I think the following is true?
$E$ compact $\iff$ $E$ has Heine-Borel property
$E$ countably compact $\implies$ $E$ compact (Does the reverse hold?)
But are the definitions of Heine Borel and Countable Compactness really saying the same thing? It occured to be that Heine-Borel could be a stronger statement than countable compactness because the cover for $E$ in Heine-Borel could be a uncountable union of open sets.
Could anyone clarify my doubts?
Best Answer
What you call Heine-Borel is just called "compact" in general topology.
Countably compact is the same in general topology (every countable open cover has a finite subcover).
There also is a notion of sequential compactness (every sequence has a convergent subsequence) (what you call compact).
For metric spaces all three are equivalent; for general topological spaces there is only the implication "compact implies countably compact", which is true by definition, and the notions are all different. If a metric space sequentially compact or countably compact imply that the space is separable (!) and thus second countable, allowing us to reduce an arbitrary open cover to a countable one. And as metric spaces are Hausdorff and first countable (countable local base) we see that sequentially compact and countable compact are mutually equivalent in that context.
So all three are equivalent in metric spaces for several general reasons: mostly because metric spaces fall in nice categories of spaces where sequences suffice and all countable cardinal invaraints coincide (Lindelöf is separable is second countable etc).