(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

(a) Prove thatR, with the cocountable topology, is not first countable.

(b) Find a subset A ofR(with the cocountable topology), and a point z in the closure of A such that no sequence in A converges to z.

2. Relevant equations

(The cocountable topology onRhas as its closed sets all the finite and countable subsets ofR).

3. The attempt at a solution

I'm not really sure how to prove this. I know that if I could find a set and point meeting the conditions in part (b), then that would prove part (a), as in a first countable space X a point x belongs to the closure of a subset A of that space if and only if there is a sequence of points of A converging to x. However, I assume that I am expected to prove thatRis not first countable with this topology some other way for part (a).

Either way, I'm stuck!

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# R with the cocountable topology is not first countable

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