Original sentence: Consider  Observe that B is a directed set under inclusion and for each  for all  Consider the net  I claim that this net cannot have a convergent subnet since for each x there exists  but for all  we have that  and thus is not in some open neighborhood of x.First suppose that X is compact
Edited sentence: Consider the net.Observe that B is a directed set under inclusion and for each , there exists an  such that  for all.This net cannot have a convergent subnet because, for each x there exists  such that , but for all , we have that  and thus is not in some open neighborhood of x.{| class="toccolours collapsible collapsed" width="90%" style="text-align:left".|First, suppose that X is compact