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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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I expect this to depend a lot of which theorems you want to be able to prove. In a first-order setting where you can only speak of points and sets of points, it's not clear that one can even speak about, say, compactness, so you'd have to restrict yourself to a subset of point-set topology anyway. –  Henning Makholm Sep 23 '11 at 23:42
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Perhaps pointless topology is close to what you're imagining as a single-sorted theory? –  Henning Makholm Sep 23 '11 at 23:50
    
@Henning: Thanks for the hint (about which theorems to be able to prove). But don't you suggest that I have to expand my view on point-set topology (instead of restricting it) - to higher orders? –  Hans Stricker Sep 24 '11 at 0:01
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I'm not sure I can give you any general advice. It all depends on what you're aiming to do. It's certainly possible to write down some first-order axioms for a topological space, and reason formally about those, just like it is possible to study, say, the group axioms in algebra. But many interesting theorems in group theory cannot be formulated in the language of the group axioms, because they are about the relation between different groups -- and usually the response to that is not to look to higher order formulations but just to use set theory whenever that's more convenient. –  Henning Makholm Sep 24 '11 at 1:02
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(cont.d) Another point is that the boundaries of "point-set topology" are not really well defined. Historically all of set theory -- including notions such as cardinality -- used to be considered to be a specialty within point-set topology. (Indeed, when I studied math, the mandatory set theory in the undergraduate curriculum was in a course entitled "general topology"). So if you broaden your scope enough it's likely that you end up doing set theory anyway. And I'm not sure there is any clear place to draw the line that is objectively better than other places you could choose to draw it. –  Henning Makholm Sep 24 '11 at 1:05

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