Is injectivity or surjectivity enough to show countability

Question

I'm new into Measure theory and I'm self-taught. For some time now, I have been encountering proofs that showed countability using injectivity or surjectivity. Here are examples https://proofwiki.org/wiki/Countable_Union_of_Countable_Sets_is_Countable/Proof_1 and https://proofwiki.org/wiki/Countable_Union_of_Countable_Sets_is_Countable/Proof_2

To me, a set is countable if and only if there exists a one-one correspondence or bijection between it and the set of natural numbers, $\Bbb{N}.$

MY QUESTION

Is injectivity or surjectivity enough to show countability as the above proofs have done?

Answer 1

The usual definitions are that:

1. a set is finite if it is in bijection with $\{1 , 2, \cdots, n\}$ for some $n$
2. a set is countably infinite if it is in bijection with $\mathbb{N}$
3. a set is countable if it is finite or countably infinite

Then, the following conditions are equivalent for a set $S$:

1. $S$ is countable
2. There is a surjection $\mathbb{N} \to S$
3. There is an injection $S \to \mathbb{N}$