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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?

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    As Henning said, this is really about how you limit your topology. If you want it to be limited, you'd have to take an infinite chunk, probably close to the entire ZFC. Either way you would want to have extensionality, power set, union, infinity, foundation, choice. Should you insist on minimality perhaps there are a few pieces of replacement you can toss away. I doubt there can be a nice description of these without some further limitations in the question.2011-09-24

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