Talk:Equivalence of Definitions of Compact Topological Space

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)