6
$\begingroup$

First of all, rest assured that I have used the search tool and read Why do we require a topological space to be closed under finite intersection? and https://mathoverflow.net/questions/19152/why-is-a-topology-made-up-of-open-sets.

I'm looking for a more logical approach on this issue. I understand intuitively the reasons why an intersection of infinitely many open sets may fail to be open, but only in the case of a topology induced by a metric. In the general sense, what would be a somewhat rigorous argument? I feel that some counterexamples aren't enough for me to deeply understand the motives. I've read somewhere that the intersection of an infinite family of sets is related with an infinite conjunction, which is not allowed in the usual logical framework we use for mathematics. Is this the issue? Could you explain it in more detail? Are infinite disjunctions permissible, then? Finally, if we were to work within an infinitary logic, would this "issue" disappear, and consequently, would the building of topology become somewhat trivial, or less rich and interesting?

Thanks.

  • 3
    What do you mean by "rigorous argument"? We're talking about a *definition* (the definition of "topology"); examples and counterexamples are precisely what guide definitions. You cannot logically demonstrate a definition.2011-03-31
  • 0
    Unions are related to *disjunctions*, not conjunctions; intersections are related to *conjunctions*. This because $x\in\cup_{i\in I} A_i$ if and only if $\lor_{i\in I} x\in A_i$ is true.2011-03-31
  • 2
    I don't understand what you mean by "logical." A counterexample is a counterexample. Once you find one, you don't need to work more to find others to disprove what you're trying to disprove.2011-03-31
  • 0
    @Arturo: I'm referring to the relation I think these concepts have with infinite conjunctions. I did say "somewhat" :) While I understand that in some sense the examples and counterexamples are the fundamental motivation for the definitions, it's the relation between the definition of a topology and the logical issue of long statements what I'm particularly interested about. Of course, I'm not entirely sure about any of this, but I believe I've read mentions of this connection.2011-03-31
  • 0
    @Arturo: About the conjunction/disjunction mix-up, I'll edit.2011-03-31
  • 0
    @Qiaochu: I understand that, but I don't find a counterexample very enlightening. For example, if I claimed that $x^2 = -2$ for all real $x$, and then found that when $x$ is $3$ I have a counterexample, I would know that my claim was false, but I wouldn't be seeing the deeper motives. In this question I'm wondering about the connection, if it exists, between the size of families of sets that unions and intersections can be taken over and infinitely long logical statements.2011-03-31
  • 0
    @Abel: in that case, I don't see how the question you're asking isn't answered by the MO threads.2011-03-31
  • 1
    @Abel: What "logical issue of long statements"? An arbitrary union may be seen to correspond to an arbitrary disjunction, an arbitrary intersection to an arbitrary conjunction. But you *do* realize that one can equally well define a topology in terms of its *closed* sets, instead of its open sets? If you think "arbitrary union" is fine because it somehow "is" a disjunction and "arbitrary intersection" isn't because it somehow "is" a conjunction, then how would that square with defining topology using closed sets, where you *would* have "arbitrary intersection* but *not* "arbitrary union"?2011-03-31

3 Answers 3