While playing around with some basic general topology, I have thought of some problems whose solutions are not so obvious (at least to me), and surprisingly I do not remember having seen these anywhere.
Disclaimer #1 : I have already posted those problems elsewhere, but I was unlucky and received no answer.
Disclaimer #2 : The French and English notions of compact sets are notoriously different. I will try to write it right (and it might be actually easier in English).
So, let us take any topological space $(X, T)$. As a matter of prophylaxy, I will assume that $X$ is not empty. A subset $K$ of $X$ is said to be compact if, from any covering of $K$ by open sets one can extract a finite sub-cover.
My first question is: When are the compact subsets for the topology $T$ the closed subsets for another topology $T^*$ (modulo the whole space $X$)?
Trivially, any finite union of compact subsets is compact. If any intersection of compact subsets is compact, then the set of compact subsets plus $X$ are the closed subsets for some topology $T^*$ on $X$. One can see the open subsets of $(X,T^*)$ as the open neighborhoods of infinity for the Alexandroff compactification of $(X, T)$, without said infinity. If $(X,T)$ is Hausdorff then any intersection of compact sets is compact (this is because the compact sets are then closed). However, this condition is not necessary: with the coarse or cofinite topology, any subset is compact, so that any intersection of compact subsets is compact. Do the topological spaces for which this property holds have a name, or a characterization?
My second question is: Assuming that the property discussed above is satisfied, is there anything interesting to be said about the operation $T \to T^*$?
For instance, if $T$ has the property that any intersection of compact subsets is compact, then does $T^*$ have the same property? This would be important as it would allow us to iterate the operation $T \to T^*$. In addition, it looks like $T^*$ can be seen, in some cases, as a dual topology on $X$. If $T$ is the coarse topology on $X$, then $T^*$ is the discrete topology, and $T^{**}$ is the cofinite topology. Then the sequence stabilizes, as $T^{***}$ is again the discrete topology. In a similar way, If $T$ is the usual topology on $\mathbb{R}^d$, I think (I have not written down the argument properly) that $T=T^{**}$. Are those special cases, or is there a more general pattern?