Are All Countable Sets Closed?

[OhMyMarkov]

Hello everyone!

I want to show that all countable sets are closed. I can show that finite sets are closed, and the set of all natural numbers is closed by showing its complement to be a union of open sets. Now, can I start like this:

A is a countable set. Every element in A can be "mapped" to an element in N by the property of countability (I presume). N is finite, so A is finite too.

Is there proof correct, if it is but technically incorrect, could you suggest a better proof.

Thanks!

 

[tarantella]

$\Bbb N$ is not finite!

And not all countable sets are closed: take the real line with usual topology, and $S:=\{n^{-1},n\in\Bbb N\}$ is countable, but not closed (as $0$ is in the closure but not in the set).

 

[HallsofIvy]

Another example: the set of all rational numbers is countable but not closed- its closure is the set of all real numbers.

 

[OhMyMarkov]

I apologize about saying N is finite, I forgot to edit that out. I believe I must review what countability strictly means.

 

[Fantini]

A set is countable if it is finite or there is a bijection with $\mathbb{N}$. :D

 

[hmmmmm]

If you consider the naturals (any subset) or rationals or something with the discrete metric then these are open, so you have (at least) countably many countable sets that are open :)