6

I'm studying general topology and a question has come to my mind.

We have defined a topological property to be a property which a (viz. any) topological space can satisfy or not satisfy, and such that, if satisfied by a space, is also satisfied by every space homeomorphic to it.

I can see the ambiguity of this definition lying in its lacking to specify the language in which the properties are expressed. Anyway, I was wondering if, in some appropriate language, topological properties are enough to capture the notion of homeomorphisms.

More precisely, is it true that, if two spaces satisfy the same topological properties written in an appropriate (formal) language, then they are homeomorphic? Feel free to make assumptions on the language of the properties.

Disclaimer If we think about properties in the most general and informal sense, then the answer is yes. Indeed, given a topological space X, "being homeomorphic to X" is a topological property. As a result, given another space Y having the same topological properties as X, it is indeed homeomorphic to X. The question may get more interesting restricting the language in which properties can be expressed.

Amanda Wealth
  • 255
  • See this question: https://math.stackexchange.com/questions/1990948/if-two-topological-spaces-have-the-same-topological-properties-are-they-homeomo – User3113446 Mar 27 '24 at 19:06
  • 3
    In the generality you wrote down, for any fixed topological space $X$, “homeomorphic to $X$” would be a topological property. Clearly, such properties capture a topological space up to homeomorphism. – David Gao Mar 27 '24 at 19:06
  • @DavidGao Yes, I was just realising it and adding to the question. Read further. – Amanda Wealth Mar 27 '24 at 19:09
  • @User3113446 Thank you, but there I see no specifying of the language involved. – Amanda Wealth Mar 27 '24 at 19:12
  • 1
    @AmandaWealth It might be a bit hard to find something of this nature. After all, most interesting topological properties are written in the language of set theory, and “homeomorphic to $X$” is also expressed in that language. But perhaps by cutting off some potentially interesting properties, there is a possibility of getting something? Perhaps something like understanding topological spaces as second order structures with a single second order predicate that identifies which sets are open, then what topological properties can be written in that language? – David Gao Mar 27 '24 at 19:24
  • @DavidGao I thought this to be a somewhat classical problem, being just a notion of elementary equivalence stated in a language which is not first-order. This said, I am not enough into second-order logic to answer your question, but I guess that expressing "hoemomorphic to X" for a certain X can be possible, I am more doubtful about expressing "to be a topological space" (how do you cope with infinitary unions?). – Amanda Wealth Mar 27 '24 at 19:32
  • Maybe we can sidestep the syntactic aspect of the question by thinking in terms of the classes of topological spaces satisfied by the statements of the proposed language. So we're looking for a collection of classes (closed under complement and other logical operations?) that does not include (all? any?) homeomorphism classes (i.e. statements equivalent to "is homeomorphic to X") but such that if you know whether a given space belongs to each class, you can determine its homeomorphism class. – Karl Mar 27 '24 at 19:35
  • 1
    @AmandaWealth Yeah, that is a problem. Then again, an inability to axiomatize topological spaces does not necessarily mean you can’t consider interesting properties expressible in that language. Hausdorff property, for example, can definitely be expressed. (Or maybe just consider doing this in third order logic. But that feels like grasping at straws.) – David Gao Mar 27 '24 at 19:50

1 Answers1

3

is it true that, if two spaces satisfy the same topological properties written in an appropriate (formal) language, then they are homeomorphic?

since first order theories with infinite models always have elementarily equivalent but non-isomorphic models, one has to look for something else. hen:

if you're willing to allow for one infinitary operation, the theory of complete heyting algebras is a good first approximation: every topological space provides a model, and there are ways to express many properties one can regard as 'topological', though not all models 'are' (or rather, 'are provided by') spaces, and there are some issues involving separation properties, so that some (non-hausdorff) non-homeomorphic spaces end up being 'elementarily equivalent'

another possible approach is via a relational theory of ultrafilter convergence, which presumably can be developed in two-sorted FOL

ac15
  • 984