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Question: Does there exist some real-valued function $f(x)$ where $f: \mathbb{R} \to \mathbb{R}$, for which $\lim_{x \to \infty}$ converges on a non-Hausdorff topology but does not converge on a Hausdorff topology?

For the purposes of this post, let me first define what I take to be a Hausdorff topology:

Definition: A topology, T$_{2}$, is Hausdorff if and only if for any two points $x_{1}$ and $x_{2}$, where $x_{1} \ne x_{2}$, there exist open sets, $O_{1}$ and $O_{2}$ such that $x_{1} \in O_{1}$; $x_{2} \in O_{2}$; and $O_{1} \cap O_{2} = \oslash$ where $\oslash$ is the null set.

Example of a non-Hausdorff topology: Define a set $\varepsilon_{4}$ consisting of all the events of ordinary (3+1)-dimensional Minkowski space, $M^{4}$. Remove the set $F$ containing the spacetime event 0, and all subsequent events both inside and on the future light-cone with its vertex at 0. Replace $F$ by two copies, $F_{1}$ and $F_{2}$. The basis for such a topology on $\varepsilon_{4}$ is thus (1) any open set in [$M^{4}-F$]$\cup F_{1}$ is an open set in $\varepsilon_{4}$; (2) and open set in [$M^{4}-F$]$\cup F_{2}$ is an open set in $\varepsilon_{4}$.

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closed as not a real question by S. Carnahan, Mariano Suárez-Alvarez, Reid Barton, Charles Siegel, Pete L. Clark Feb 5 '10 at 4:55

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

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As far as I can see, your question does mot make much sense. Can yo elaborate on what you mean by 'finite', for example? –  Mariano Suárez-Alvarez Feb 4 '10 at 16:54
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@Ian, the cardinality of a set exists outside of a topology on it. Could you give some examples of "finite" objects? Can you give some examples as to what you mean? –  Tom LaGatta Feb 4 '10 at 18:04
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@Ian: it's not going to benefit anyone if you get defensive. No one is saying that you are not a competent mathematician. Rather they are saying that you simply haven't yet succeeded in communicating the meaning of your question to us. It sounds like you are saying that you want to change the cardinality of a set by changing the topology that is placed on it. This is certainly not possible, so it can't be what you really mean to ask. What is it that you really mean to ask, in mathematical terms? (You are asking a math question, and not a physics or philosophy question?) –  Pete L. Clark Feb 4 '10 at 20:09
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Ian, that's completely different from the question you asked. –  Qiaochu Yuan Feb 5 '10 at 1:12
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I have closed the question. This is not intended as a judgment that it was intrinsically valueless, but rather that it simply wasn't clear as a mathematical question, and the sequence of comments was not making substantial progress towards clarification. I invite you to try again. Suggestions: 1) stick entirely to mathematics. 2) make sure you use terminology in a standard way. 3) proofread your question [the current version defines a Hausdorff space -- which seems unnecessary, BTW -- but the definition given is incorrect]). –  Pete L. Clark Feb 5 '10 at 5:02

2 Answers 2

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.
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Ian gives a very clear statement of his underlying concerns in this blog post of his: quantummoxie.wordpress.com/2010/01/31/the-nature-of-mathematics . If he posted it as a separate (soft, community wiki) question I think people would be happy to answer. –  Qiaochu Yuan Feb 5 '10 at 5:01

Note: this is a response to an earlier version of the question, and so is rather speculative. I am not sure that I can edit this to be much use in response to the newer version, so I'm leaving it as it stands for now, except to strikethrough a closing remark which was wide of the mark.


I think that the original questioner may have something like this in mind (though since I'm not a telepath, Jim, corrections/comments are welcome).

When we explain the definition of Hausdorff, we talk about separating points from other points using open neighbourhoods. Now if your intuition comes from metric spaces, these neighbourhoods are balls. So one might think that Hausdorffness is to do with separating things by balls, Furthermore, one might be thinking of some notion of minimal ball size -- this isn't what Hausdorffness is about, but bear with me! I'm trying to recreate a train of thought, not recapitulate the correct definitions -- and so get the idea that in "contexts which are Hausdorff" certain postulated objects -- the collection of all Widgets that satisfy the Sveshnikov-Pelikan equation - are forced to be finite because of "the need to separate constituent parts with balls". (See the original post's 4th para.)

Pursuing this train of thought, one might then wonder that if this postulated object is not finite, this is something to do with the failure of Hausdorffness. Again, I think this comes from a misapprehension about the Hausdorff separation condition; but at least this interpretation makes some sense of the original post's 1st para.

For the record: of course, things are more likely to be Hausdorff when you have more open sets, and of course a completely discrete space is Hausdorff for the trivial reason that every set is open. At the other extreme, a space with the indiscrete topology and more than two points has no chance of being Hausdorff. However, this has absolutely nothing to do with constraining the underlying set to be finite or infinite, contra the apparent guess of the original question. (If it helps: in the definition of Hausdorff, we don't constrain "the size of our open balls" before picking our two points; given any two points in, say, a metric space, we then have the freedom to choose a mesh size which will distinguish between them.) In particular, the sentence which starts "The topology of the multiverse would therefore be non-Hausdorff..." is in my view based on a misunderstanding, either of the word "Hausdorff" or the word "therefore"...

A comment for Ian, if I may: the reaction you got was because you started with a misuse of mathematical terminology, inserted into a question with words like "context" which are by nature philosophical/cultural rather than mathematical. It's that clash of tones which I think confused/irked some of the commenters.

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Actually, I have thought of one context where a set, defined a certain way, will be "finite in the Hausdorff context but infinite in the non-Hausdorff one" -- however, I don't think it has any physical reality and will not help the original poster. The context is "closure of 0 " in a complete seminormed space -- just look at 21 given the quotient topology for an example where this closure is infinite; and look at your favourite Banach space (e.g. the ground field) for an example where it is finite. –  Yemon Choi Feb 4 '10 at 19:29
    
@YC: The one mathematical thought that I got from the question was an example of a space (e.g. a pseudometric space or uniform space) which is infinite, but whose universal separated (T_0 = T_1 = T_2 for uniform spaces) quotient is finite, e.g. the indiscrete topology. Does this have something to do with the question? With your answer? (I am very confused...) –  Pete L. Clark Feb 4 '10 at 20:03
    
@PLC: well, that's certainly related to my comment, but I'm not sure if it's connected to the question. I'm certainly not sure that these kinds of issue help with the questioner's underlying motive (which seems to be seeking some kind of framework to reconcile physical reality with multiverses) –  Yemon Choi Feb 4 '10 at 20:32
    
Oops, I see that in my first comment some of the LaTeX went wrong. The example I meant to give was l^2(N) / l^1(N), given the quotient topology –  Yemon Choi Feb 4 '10 at 21:32
    
@YC: Actually I had originally said "on" and not "in the context of" but changed it when the initial complaints came in. Dinner is on the table but I will be back in a bit to clarify the original question. –  Ian Durham Feb 4 '10 at 23:40

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