# Talk:Equivalence of Definitions of Compact Topological Space

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As reported by private email to the ProofWiki admin (corrected for grammar):

- "The following section contains a false claim in a purported proof:

- (4) $\implies$ (3)
- Let $\mathcal F$ be a filter on X.

- As we have that Every Filter is Contained in an Ultrafilter, there exists an ultrafilter $\mathcal F'$ such that $\mathcal F \subseteq \mathcal F'$..

- By (4) we know that $\mathcal F'$ converges to a certain $x \in X$.

- This implies that x is a limit point of $\mathcal F$.

- NO IT DOESN'T!!

- $x$ is a limit point of $\mathcal F$ iff the neighborhood filter of x is coarser than $\mathcal F$ (see: Definition:Convergent Filter). That filter can be coarser than $\mathcal F'$ without being coarser than $\mathcal F$. This is just a non-sequitur." --Scott Engles

This will be attended to in due course. --prime mover 22:40, 13 July 2012 (UTC)

- It appears, based on the link to the result about superfilters, that this has been attended to, so I deleted the questionable template. Is there anything questionable remaining? --Dfeuer (talk) 06:21, 2 January 2013 (UTC)

- Probably not. The above flame was received by email from one of our typical detractors. --prime mover (talk) 06:23, 2 January 2013 (UTC)