When is a Collection of Finite or Countable Subsets a Topology?

[jacobrhcp]

[SOLVED] very basic topology questions

Homework Statement

Let X be a set and T be the collection of X and all finite subsets of X. When is T a topology? Let T' be the collection of X and all countable subsets of X, when is T' a topology?

The Attempt at a Solution

it's clear the empty set and X are in T
if two finite subsets united, the new set is also a finite subset
the intersection between two finite subsets is again finite.

The only hole I can find is when I unite an infinite amount of finite substes of X, but what restriction does that give on X? Surely, if X is finite T is a topology... but surely I can say a bit more than that?

I use the same reasoning for the countable version of the question, and I find that if X is countable T' is a topology... but again, I'm left wondering if there is nothing more to say.

 

[quasar987]

Not really!

 

[jacobrhcp]

what do you mean?

That I said all there is to say?

so are there no infinite sets for which the collection of finite subsets (and X itself) form a topology?

 

[HallsofIvy]

Yes, if X is finite then T is a topology. If, however, X is not finite, choose anyone "open" set. Call it Y. Now take the union of all "open" sets except Y. Can you show that the union is not finites and so not an open set? That proves that T is a topology if and only if X is finite.

For the second part, you can say T' is a topology if and only if X is countable itself.

 

[jacobrhcp]

ah, thanks... I'm confident I can do that :)