You're probably already thinking "silly person" but hear me out.
Compactness: Every open cover (of a set $A$) has a finite subcover (this means every cover $\{U_\alpha\}_{\alpha\in I}$ where $A\subset\bigcup_{\alpha\in I}U_\alpha$ or $A=\bigcup_{\alpha\in I}U_\alpha$ where the $U_\alpha$ are open with respect to the subset of topology)
This means EVERY open cover, there's a finite bunch of open sets which cover it. So it's sort of like size of a set, it's "bounded" because you can always cover it finitely. I like that.
But then consider the set $(0,1)$ (or any open interval) on $\mathbb{R}$ this is bounded, it isn't a "big" set, but it isn't compact (yet [0,1] is)
So my question is what does compactness actually try and do? "Oh great the set is compact" what does that mean?
Can someone describe it qualitatively
By the way I have read the wikipedia page, I've done some research. Compactness seems to only be useful for some proves but even then the proofs are not "and with this property we can do this" it's more like a "clever trick"
