Seeing as the comment thread to the original question is running out of control, let me just record some attempts to formulate a question which might (a) be related to what Ian Durham is asking, and (b) is more palatable to some of the people, myself included, who find the original question hard to answer meaningfully.
First of all: I guess we are taking as a working principle
... it is impossible to simultaneously have an infinite number of physical objects of non-zero size in the universe
The example given of an object in the original question is something like "a quantum channel" - now since I'm a physics ignoramus I don't know what the ontological status is of such a beast, but let's suppose for sake of discussion that it does have "size" and that therefore only finitely many of such can exist in a given physical system. This is presumably some argument about physical observables being quantized, but someone else is welcome to correct me on this.
Secondly: there are constructions in mathematical physics which seem to be of an infinite nature. The example given seems to be a "potential infinity", i.e. what are we approaching if we tensor a channel with itself repeatedly.
Now, my interpretation of what Ian may be trying to ask -- and I have to say, in my personal opinion I've not found it at all easy to discern what his underlying question is -- goes like this:
(i) are there contexts in "mainstream abstract mathematics" where an implicitly defined "object" -- such as, the solution space of some differential equation, the solution set of some algebraic equation, the set of accumulation points of some sequence -- which depends on some outside flavour (choice of ground-field for an algebraic equation; choice of topology on some ambient space which reasonably admits more than one topology; an ambient topos in which the construction is supposed to live), might have finite cardinality for one choice of flavour, but infinite cardinality for other choices?
(ii) does this have anything to do with whether we equip a given space, broadly and vaguely conceived, with a Hausdorff or a non-Hausdorff topology?
(iii) do either of these have any connection to the original subject, namely that certain mathematical constructions appear to have physical meaning yet be defined in terms of unphysical infinities?
The answer to (i) is in my view "yes, but so what?" and the answer to (ii) is in my view "I don't really think so". Moreover, I don't think (iii) is really dependent on (ii), and so my overall impression is that the "Hausdorff discussion" is a red herring.
Lastly, I am having difficulty making sense of the reasoning behind this sentence in the original question:
Now suppose that one of the various branching spacetime interpretations of quantum mechanics (MWI, MMI, etc.) is correct (personal aside: I am agnostic on this issue). The topology of the multiverse would thus be non-Hausdorff and, given these interpretations of QM, there ought to be an infinite number of branches. Given that, an infinite physical realization becomes possible.