Well it comes down to word-play again. I'm confused to the core of my bones as to why the following isn't equivalent to saying that a space is compact
Every open cover is finite.
A compact set must have every open cover in which there is a finite subcover.
Well, the statement tells me that every single possible open cover is finite. So...why doesn't this qualify?
Subcover needn't be a proper subset/cover so if I have every open cover being finite...the open cover itself can act as a subcover(which is of course finite) so voila, we're done..no?
Elaborate explanation much appreciated on this