0
$\begingroup$

I have some problems with the arbitrary union and finite intersections of sets.

Let C be the set of finite unions of subsets of $\mathbb{N}$ of the form $$ \left( n \right) = \left\{ {n,2n,3n,...} \right\} $$ and the empty set.

The problem is to show that the sets of this form are the closed sets of a topology on the set of natural numbers.

  • 2
    Unrelated question: why do I often see "it´s" in place of "is"? Is it just a weird typo?2011-08-17
  • 4
    @Daniel: A topological space consists of a set $X$ and a family $\tau$ of subsets of $X$ that contains $\emptyset$, $X$, and is closed under arbitrary unions and finite intersections. What is your set $X$ here, and is $n$ an arbitrary integer, an arbitrary positive integer, an arbitrary natural number, or something else?2011-08-17
  • 0
    When you write $n$, do you mean $\{0,\ldots,n-1\}$? What about $2n$ and so on?2011-08-17
  • 0
    @Asaf: No, I think he's talking about finite unions of sets of the form "all (positive integer) multiples of $k$". So the sets will be "all (positive integer) multiples of one of $n_1,n_2,n_3,\ldots,n_k$" for a finite collection of $n_i$s. My question is what can the $n_i$ be, and what is the underlying set supposed to be.2011-08-17
  • 4
    I don't understand the question for a different reason than nitpicking. As I understand your question you want to check if your sets form the closed sets of a topology, so you should not have to consider *arbitrary* unions but only finite ones. On the other hand, you must check that arbitrary *intersections* are of the desired form.2011-08-17
  • 0
    @Theo: Ah, good point; *I* at least missed the "closed sets".2011-08-17
  • 0
    I mean all "n" over the natural numbers, and remember these sets are closed, and not open2011-08-17
  • 1
    @Daniel: And what is your $X$? And, as Theo points out: if these are meant to be the closed sets, why are you worried about arbitrary unions? You don't need to consider arbitrary unions to check if a family of subsets are the closed sets in a topology.2011-08-17

1 Answers 1