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Limits first, then topology.

  1. Aug 22, 2011 #1

    Fredrik

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    In at least one book and one Wikipedia article, I've seen someone specify which sequences are to be considered convergent, and what their limits are, and then claim that this specification defines a topology. I'm assuming that this is a standard way to define a topology. I want to make sure that I understand it.

    Some of my thoughts: Suppose that we somehow specify all the convergent sequences in a set X, along with their limits, in a way that ensures that all subsequences of a convergent sequence have the same limit. Let F be the set of sequentially closed subsets of X. Then it's easy to show that the following holds: [itex]\emptyset,X\in F[/itex]. Every intersection of members of F is in F. Every finite union of members of F are in F. (We need the requirement about subsequences to prove that last one). So now we define [itex]\tau[/itex] as the set of subsets of X whose complements are in F, and use de Morgan's laws to show that it's a topology. Then we show that if the original specification says that [itex]x_n\rightarrow x[/itex], then every open neighborhood of x contains all but a finite number of terms of the sequence.

    Looks good so far, but then I noticed that many different specifications give us the discrete topology, and that when we go in the other direction (i.e. determine all the sequences that are convergent with respect to a given topology), the cofinite topology gives us one of those specifications. Since the cofinite topology is a subset of the discrete topology, I'm thinking that my idea gives us the largest topology that ensures that all the sequences that were specified as convergent, are convergent with respect to the topology. Maybe we want the smallest?
     
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  3. Aug 22, 2011 #2

    Landau

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    A topology is uniquely determined by its convergent nets; in general sequences do not suffice.

    So yes, if you start with a collection sequences with a specified limits, and demand that these are all the convergent nets, then you get a topology whose closed sets are the sequentially closed ones.

    Could you elaborate on your last paragraph? I don't really understand what you say about the discrete and cofinite topology. The largest (assuming you mean largest = finest = strongest, so large in the sense of set inclusion) topology making certain sequences converge is the discrete topology, it makes every net converge to every point.
     
  4. Aug 22, 2011 #3

    Fredrik

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    I meant that if [itex]\tau[/itex] and [itex]\tau'[/itex] are topologies on the same set and [itex]\tau\subset\tau'[/itex], then [itex]\tau'[/itex] is "larger". I guess that's what most books call "finer". The trivial/indiscrete topology makes every point a limit of every sequence. Every topology that makes singleton sets open gives us the same convergent sequences as the cofinite topology: the ones such that all but a finite number of terms are the same. The cofinite topology and the discrete topology both have this property.

    Consider a few different specifications of "all convergent nets and their limits":

    1. No nets are convergent. This makes every set closed (if we define "closed" the way I suggested in post #1), and that makes every set open. This is the discrete topology.

    2. The only convergent nets are the constant sequences, and that for each x, the limit of x,x,x,... is x. Every set is closed, so that's the discrete topology again.

    3. The only convergent nets are the sequences such that all but a finite number of terms are equal to a specific value, and that value is the limit of the sequence. Every set is closed, so that's the discrete topology again.

    On the other hand, if we start with the discrete topology, and define limits (of sequences) using its open sets, as I said above, the convergent sequences will be the ones mentioned in #3 above. And we get the same result if we start with the cofinite topology.

    Hm, maybe this gets a bit weird because I've been focusing too much on sequences (rather than nets). I will have to think about this.

    It looks like I remembered things wrong when I mentioned a book that specified topologies by specifying the convergent sequences. It was a book that I had only quickly looked at. I just looked again, and he's talking about nets, not sequences. Here. Let's find that Wikipedia article...OK, there it is. It's talking about sequences, so at least I remembered something right.
     
  5. Aug 22, 2011 #4

    micromass

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    Hi Frederik!!

    You touch some very interesting points in your post.

    First, in general, sequences do not suffice to know the topology. For example, the discrete topology and the cocountable (not the cofinite!) topology have the same sequential limits, i.e. the eventually constant sequences.

    However, knowing all the convergence points of filters (or equivallently nets) IS sufficient to know the topology. This leads (for example) to the study of limit spaces and pseudotopological spaces. If you want, I'll look up the axioms that filter-limits need to satisfy in order for it to be a topological space.

    Anyway, and now comes the interesting part of my post, spaces which are defined by sequences are called sequential spaces. And as it turns out, sequential spaces are concrete coreflective in Top (forget this). All that this means is that for every topological space [itex](X,\mathcal{T})[/itex], there is a finest sequential space that has the same sequential limits. How do we construct this: well, define a sequential closed set as a set which contains all the limit points of its sequences. Take these sets as the closed sets of your topology. This defines the finest sequential space that satisfies the properties.

    It can be proven that these does not exist a coarsest topology (or a smallest topology) that satisfies our properties.

    Finally, the wikipedia article strikes me as wrong. We can not define a topological space by defining its limits. It would be more honest to define the topology by the family of semi-norms [itex]p_{K,\alpha}(\varphi)=\sup_{x\in K}{D^\alpha \varphi}[/itex]. I think this is what the writers of that page wanted.
     
  6. Aug 22, 2011 #5

    Landau

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    Yes, that's what I meant as well (larger in the sense of inclusion).
    I am sorry, that is of course what I meant (I wrote mistakinly 'discrete' instead of 'trivial'). The finer=larger=stronger the topology, the less convergent nets.
    I think I understand what you are saying. You seem to be bothered by the fact that the procedures 'convergent sequences->topology' and 'topology->convergent sequences' are not each other's inverse. And as you realized (and as I mentioned), this is because you need to consider nets or filters instead of sequences.

    I agree with micromass: I think wiki only tries to say that one gets the structure of a locally convex space via that collection of seminorms {p_i}; this means that the topology is the initial topology w.r.t. the maps [tex]\{x\mapsto p_i(x-a)\}_{i,a}.[/tex]
     
  7. Aug 22, 2011 #6

    Fredrik

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    Yes, I think I understand now that a specification of all sequences that we're going to call "convergent" and the points we're going to call their "limits", doesn't single out exactly one topology from which the original specification can be recovered. But it seems to me that since any intersection of topologies is a topology, there should be a unique smallest topology from which the original specification can be recovered.

    Phrases like "know the topology" suggests that you're thinking about a slightly different problem than what I have in mind. You seem to be saying that knowing all the nets that are convergent with respect to some topology, and knowing their limits, is sufficient to find that topology. But what I'm curious about is what happens if we just specify a bunch of nets and call them "convergent", and then specify a bunch of points and call them the "limits" of those nets. I tried to do that with sequences. It seems that not all specifications give us topologies, but if we add the requirement that the specification must be such that all subsequences of a convergent sequence have the same limit, then we end up with a topology.

    I haven't had time yet to see what happens if I start with nets instead, but I'll start thinking about it now. It's not immediately obvious to me that every reasonable specification defines a topology. Looks like I'll have to think of something like the requirement about subsequences, but that makes sense for nets. Hm, I have never heard the term "subnet" (other than in the context of IP addresses, and the TV show 24, where computer nerds use it in almost every sentence even though the writers clearly don't know what it means).
     
  8. Aug 22, 2011 #7

    quasar987

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    Suppose you give a condition that characterizes the convergent sequences in a set X. In general, this does not specify a topology on X uniquely, as you observed. But, if there exists a metrizable topology whose convergent sequences are characterized by that condition, then it is unique (in the set of metrizable topologies) with that property. (Easy to show!)

    For instance, for X,Y two top. manifolds, the Whitney weak (aka compact-open) topology on C(X,Y) can be specified without ambiguity (although perhaps not very informatively!) as the unique metrizable topology such that f_n-->f iff f_n-->f uniformly on compact sets. In principle, one still has to show that such a topology exists though!
     
  9. Aug 22, 2011 #8

    micromass

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    The problem is that there might not exist a topology with the original specification. Then we take the intersection of the empty set.
    Furthermore, if there does exist such a topology, then it is not clear to me that the intersection also has the original requirement. I.e. let [itex]\{\mathcal{T}_i~\vert~i\in I\}[/itex] be the topology such that the convergent sequences are exactly specified, then it is not clear to me that [itex]\bigcap_{i\in I}\mathcal{T}_i[/itex] also has exactly those convergent sequences. There might be more convergent sequences!!
    I'll think about this for a bit more.

    Let me make precise what I mean. Let X be a set and let

    lim:{nets in X} ---> {subsets of X}

    be a function (which intuitively sends each net to its limit points). There exists a unique topological space with the previous function as the limit of nets, if and only if the following conditions are satisfied:

    • If [itex](x_\alpha)[/itex] is a net that is eventually constantly z, then [itex]x_\alpha\rightarrow z[/itex]
    • If [itex](y_\beta)[/itex] is a subnet of [itex](x_\alpha)[/itex] and if [itex]x_\alpha\rightarrow z[/itex], then [itex]y_\beta\rightarrow z[/itex].
    • Most difficult, the iterated net condition: Let [itex](y_\delta~\vert~\delta \in D)[/itex] be a net converging to z. For each [itex]\delta\in D[/itex], let [itex](x_\epsilon^\delta~\vert~\epsilon E_\delta)[/itex] converge to [itex]y_\delta[/itex], then the net [itex](x_{f(\delta)}^\delta~\vert~(\delta,f)\in D\times \prod E_\delta)[/itex] is a net converging to z.

    So if you specify the convergent nets and their limit points, then these conditions specify a topological space iff the previous conditions are satisfied.
     
  10. Aug 22, 2011 #9

    Landau

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    A subnet is not so hard to define, although there are slightly different ways which give the same nice properties: a set is compact iff every net in it has a convergent subnet, a net converges to x iff every subnet converges to x, and the like.
    I was just watching the last hours of Day 6 of 24, I think I noticed the term sometime: ) I suspect most of the technical talk is nonsense, but it is pronounced fast enough to be somewhat believable...
     
  11. Aug 25, 2011 #10

    Fredrik

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    OK. This follows from the fact that in a metric space "closed=sequentially closed".

    This is awesome. Exactly the sort of theorem I thought should exist. I suspected that I would have to include those first two conditions, but the third had not occurred to me.

    I find it tolerable most of the time, but it gets annoying at times. The worst one is in episode 4x01. Lucas Haas is stealing software from Adobe and Microsoft when a window opens on his screen (not sure if he did anything to open it), and quickly fills up with lines that look roughly like this:

    223h x CF8A BA882A85 2F86A513

    The computer is making a weird noise when the scrolling is going on. Something like a clicking sound every 1/10 second, and a little beep every 1/3 second.

    Lucas Haas: What the hell is this? Whoa. Melanie, come look at this.

    Melanie looks at it for no more than 3 seconds and says: Looks like someone's trying to corrupt the internet.

    Lucas Haas: Yeah, but it hasn't started propagating yet. They're just placing the nodes. Look at this! This thing could tank every system in the world!
     
    Last edited: Aug 25, 2011
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