# Talk:Countably Compact Metric Space is Compact

This is one of those pages where there is a different number of only-if proofs than if proofs. Rather than repeat the if proof twice, once for proof 1 and once for proof 2, it is better to separate out the two halves as separate things to be proved.

Fortunately not much needs to be done here: the if proof stands alone for all topological spaces, while the only-if is the bit which is relevant to metric spaces only.

So I recommend we lose the "iff" in the statement of the proof, and just report this as: "Let M be a metric space. Let M be countably compact. Then M is compact." This will then have 2 (or n) proofs.

We can then put another page together called "Metric Space is Countably Compact iff Compact" which will have 2 (one-line) sections: Necessary Condition: Countably Compact Metric Space is Compact, and Sufficient Condition: Compact Space is Countably Compact.

See what was done with the maze of complexity of the Heine-Borel Theorem/Real Line for a precedent for this approach. --prime mover (talk) 22:12, 31 October 2012 (UTC)

- Sounds good. While at it, let us ensure that all pages are endowed with accurate, descriptive titles. --Lord_Farin (talk) 22:15, 31 October 2012 (UTC)