TITLE: Can compact sets completey determine a topology?
QUESTION [17 upvotes]: Suppose that $\tau_1$ and $\tau_2$ are two topologies on a set $X$ with the property that $K\subset X$ is compact with respect to $\tau_1$ if and only if $K$ is compact with respect to $\tau_2$. Then is this enough information to determine whether $\tau_1 = \tau_2$?
If not (which is suspect to be the case) is there a nice counterexample, and what is the minimum amount of extra information required for $\tau_1 = \tau_2$?
I feel like there would be an issue regarding 'points at infinity.'

REPLY [3 votes]: Consider a finite set $X$,then every subset $A$ of $X$ is compact in the trivial topology $\{\emptyset,X\}$, but every subset $A$ of $X$ is also compact in the discrete topology on X (the topology where all subsets of X are considered open). If $|X|>1$, then these topologies are obviously not equal. 
I will return to your second question if it happens that I have anything useful to say about it.