I've read brief topology part of my analysis book and came across the page of concept called compactness.
I've understood at least the definition of a set being compact (One shows $K$ is compact subset of space $X$ if for every open cover of $K$ there must be subcover of it which is finitely indexed), could prove some propositions like:
"Any Compact set of metric spaces are closed",
"If any finite subcollections of collection of compact sets $\{K_q\}$ is nonempty then intersection of all its elements is nonempty".
Thought I was going in right way and however, I really wasn't at all. I know there are various proofs that tells $[0,1]$ is compact set and they are rigorously right.
But in order hand my intuition tells that :
for any $N$ positive integer, $\{(-\frac1N,\frac1N),(0,\frac2N),(\frac1N,\frac3N),...,(1-\frac1N,1+\frac1N)\}$ covers our interval fine and it is infinite but one really can't expect any subset of this to be covering $[0,1]$.
If we remove $(\frac kN,k+\frac 2N)$ (where $k=0,1,\dots ,n-1$) from the list, it clearly won't cover point $k+\frac 1N$ of $[0,1]$ and no need to talk about when it is reduced to finite size. What am I missing here?