Requirement on being countable

Question

Is this statement true?

If $A$ is in a 1-1 correspondence (bijection) with a countable set (not necessarily the set of positive integers) then $A$ is countable.

Thanks.

Answer 1

Yes that is true, by definition of countable set.

If $f:A\to B$ is bijection where $B$ is countable and $g:B\to \mathbb{N}$ is also a bijection (it exist since $B$ is countable), then is also $g\circ f :A\to \mathbb{N}$ a bijection, so $A$ is countable.