In the standard second-order, but single-sorted setting of point-set topology one has a base set $X$ and the property of being open on its powerset $P$ obeying the usual axioms. Proofs in point-set topology rely on these axioms and on the fact that $P$ is the powerset of $X$, such that set theory can be applied.
Which subset of the ZFC axioms is effectively necessary in this standard setting to be able to prove the theorems of point-set topology?
There is also a first-order, but two-sorted setting – unaware of subsets – with an extra relation (call it $\in$) in which analogues of the set theoretic axioms must be explicitly stated.
Which subset of the ZFC axioms (relating the two sorts) has to be stated in this two-sorted setting to be able to prove the theorems of point-set topology?
Finally there is – assumably – a genuinely first-order single-sorted theory (knowing only the subsets, not the base set $X$) with one or more extra relations (call one of them $\subseteq$).
Which extra relations have to be assumed and which subset of the ZFC axioms (relating these relations) have to be stated in this one-sorted setting to be able to prove the theorems of point-set topology?