- #1
jacobrhcp
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[SOLVED] very basic topology questions
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?
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.
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.