I would like to know if compactness is equivalent to limit point compactness in general for any topology. If so I'd like to see a proof, other wise I'd like to see a counter example. I am curious about this because I want to get an intuition for compactness, limit point compactness makes intuitive sense to me as a way of determining if you can "escape" the space by getting ever closer to a point "outside" the space, however the idea of open covers and open sub covers seem to come out of no where, I'd like to see an example where there 2 definitions differ so that hopefully I can see the need for the compactness definition that we use.
This is not a duplicate of limit point compactness implies compactness as I am not just asking about the first direction of implication but rather both directions of implication, I also want to know if compactness implies limit point compactness.
Let $\tau$ be a topology on ${\bf{N}}$ generated by $\{1,2\},\{3,4\},\{5,6\},...$
$\tau$ is limit point compact.
$\tau$ is not compact by considering the open cover $\{\{1,2\},\{3,4\},\{5,6\},...\}$.
A proof that $\tau$ is limit point compact. Say, an infinite set $A$. If $A$ contains an even number $k$, then we claim that $k-1$ is a limit point for $A$: The basic open set $\{k-1,k\}$ contains $k$, and $\{k-1,k\}-\{k\}\cap A=\{k-1\}\cap A=\{k-1\}\ne\emptyset$.
Similarly, if $A$ contains a odd number $l$, then $l+1$ is a limit point for $A$.