Let me preface by saying I know very little about cosmology but have been wanting to learn. I'm a mathematician specializing in hyperbolic $3$-manifolds, and am aware of some of the applications of that theory to cosmology but have yet to study the details. So please forgive any ignorance in my question, and be aware that I'm looking for help in starting to think about these concepts.
The text below has been edited to be more specific. Thanks to @ACuriousMind for clarifying some semantic aspects and pointing me to some information.
If I understand correctly, experiments have shown that, as far away from us as we can measure into space, things (e.g. galaxies) seem to be getting further apart. "Hubble's Law" gives us one version of that statement, and it seems to have remained true as we increase the distance into space by which we can measure. Skipping details, this would encompass the redshift surveys, cosmic microwave background experiments, and other studies. While these were different ways of supporting the "cosmic inflation" idea, I think their means of doing so are similar (please correct me if I'm wrong): they show the distance between things generally increasing in a sphere around our vantage point.
My question is not about those experiments, but about how it implies the entire universe is expanding. What reason do we have to think that the portion of space we have measured is representative of the whole thing? From the perspective of dynamical systems, it is perfectly natural for there to be attracting points and repelling points within the same system. How do we reject the hypothesis that we are just near a (relatively, for us) massive repelling point?
An important aspect of this question is semantic. Firstly, what do I mean by "the universe." I think my question remains valid even if I don't mean, quite vaguely, everything that exists. Physics has a notion of "the observable universe," which notably is not the same as the observed universe. Rather, it is the portion which we are theoretically capable of observing. But if I look this up on Wikipedia, the definition of the "observable universe" evokes the assumption that the universe is expanding. To a mathematician, it seems a bad idea to assume a property in the definition, that we have created the definition to study! For all practical purposes, perhaps it doesn't matter. It is probably true that we will never observe anything beyond "the observable universe," but to a cosmologist it should be relevant how far we can extrapolate a grander structure.
To move on from the semantics, let's say in this question "the universe" is the portion of reality that we are path-connected to in the topological sense. Even that bears unfounded assumptions but maybe it's good enough for this conversation.
Let me now say a bit about how small the part of the universe we've measured is likely to be, from a topologist's perspective. Consider our experiments involving curvature of space. More and more the results have been claiming "the universe is probably flat." Well, a fundamental property of a manifold is that when you measure a tiny portion of it, it seems to be flat. So either it isn't flat and we only think it is because we can barely see any of it. Or it is flat, but then it probably is infinite, in which case it is literally impossible to measure more than 0% of it.
So... what do we have to go on really when asserting anything about the universe in a cosmological sense? (From here, the Big Bang is highly suspect too.) I say this not to be snide, but because I really want to know. I especially would love to read some expository papers that would be accessible to a $3$-manifold topologist.