Can a topology be recovered from its collection of pre-compact sets?

Question

Say we are given a set $X$ along with a collection $\mathcal{A} \subseteq 2^X$ of subsets. Is there a topology $\tau$ whose pre-compact sets are exactly $\mathcal{A}$? Is that topology unique?

Answer 1

Clearly $\mathscr{A}$ must contain all finite subsets of $X$ and be closed under finite unions. It must also be closed under subsets, since if $A\subseteq B$, and $\operatorname{cl}B$ is compact, then $\operatorname{cl}A$ is compact as well. In particular, if $X\in\mathscr{A}$, then $\mathscr{A}=\wp(X)$. All of this makes $\mathscr{A}$ an ideal in $\wp(X)$, proper iff $X\notin\mathscr{A}$, i.e., iff $X$ is compact. It’s not yet clear to me whether this condition is sufficient, though I suspect that it is not.

There need not be a unique topology on $X$ making $\mathscr{A}$ the family of relatively compact subsets of $X$. Let $n$ be an arbitrary positive integer, and let $X_n=n\times(\omega+1)$, where $n$ and $\omega+1$ have their usual order topologies. If $A\subseteq X_n$, let

$$F=\big\{k<n:A\cap\big(\{k\}\times(\omega+1)\big)\text{ is infinite}\big\}\;;$$

then

$$\operatorname{cl}A=\big(F\times(\omega+1)\big)\cup\underbrace{\Big(A\cap\big((n\setminus F)\times(\omega+1)\big)\Big)}_{\text{finite}}$$

is compact. Thus, if $X$ is countably infinite, for each $n\in\Bbb Z^+$ there is a compact Hausdorff topology on $X$ with exactly $n$ limit points such that every subset of $X$ is relatively compact.

If instead we let $X_n=n\times\omega$, where $n$ has the discrete topology and $\omega$ the cofinite topology, we get a compact $T_1$-space in which every subset is compact and the largest number of non-empty compact open sets is $n$, so the topology isn’t necessarily unique even if we require $\mathscr{A}$ to be the family of compact subsets of $X$.

For yet another example, let $\mathscr{A}$ be the family of finite subsets of $\omega+1$, let $\mathscr{U}$ be any free ultrafilter on $\omega$, and let

$$\tau_\mathscr{U}=\wp(\omega)\cup\big\{\{\omega\}\cup U:U\in\mathscr{U}\big\}\;;$$

then $\tau_\mathscr{U}$ is a Hausdorff topology on $\omega+1$ in which the only (relatively) compact sets are the finite ones. Clearly the discrete topology is another topology with the same (relatively) compact sets.