Talk:Continuous Image of Compact Space is Compact/Corollary 3/Proof 2

To abcxyz: The "why?" that you appended to this proof in a "Questionable" template is in relation to a proof which you posted up yourself. This dates from the (pre-refactor) edit of the Continuous Image of Compact Space is Compact page on 19th March 2012. --prime mover (talk) 06:23, 28 November 2012 (UTC)


 * Yeah, sorry about that. I didn't remember. The only justification that I know of uses the axiom of countable choice, and I'm not sure it can be done without it. Now that I look at it, this proof doesn't add anything to Proof 1. I guess I was a bit careless then. --abcxyz (talk) 00:06, 30 November 2012 (UTC)

Unrelated question: Is this distinct from the Extreme Value Theorem in some way that I'm missing (other than one being defined on a general normed vector space and the other on $\R$)? Either way, should there be a merge/link here? --Alec (talk) 03:33, 29 November 2012 (UTC)


 * I hadn't noticed the similarity before, but the fact that one of them is on a normed vector space is a significant difference in scope, IMO. So we won't want to merge, but what we probably do want to do is to add a further proof to this result based on that Extreme Value Theorem. --prime mover (talk) 06:08, 29 November 2012 (UTC)


 * Apparently, some people think that the "Extreme Value Theorem" is a named theorem of real analysis that refers to the special case of this result when $S$ is a closed real interval. --abcxyz (talk) 00:10, 30 November 2012 (UTC)


 * Is it not, then? --prime mover (talk) 06:20, 30 November 2012 (UTC)