Nets and sequences in countable sets.

Question

Assume $A$ is topological space and $A$ is countable, the for any net $\left\{ x_{\alpha}\right\} \subset A$, can we conclude that $\left\{ x_{\alpha}\right\}$ is actually a sequence in $A$?

Answer 1

No, a sequence is a net with index set $\Bbb N$ in its standard order, the space in which we work is irrelevant. A net can assume two values and not be a sequence, etc.

That is why sequences can have subnets that are not (sub)sequences, e.g.

Answer 2

No.

A "sequence" is a function from $\mathbb{N}$ to a topological space $X.$

A "net" is a function from a directed set $\theta$ to a topological space $X.$

So you can see that any sequence is a net. but there are nets which are not sequences.

for ex: consider $f:[0,1] \to [0,1]$ where $f(x)=x$ is a net. but it is not a sequence.

Answer 3

A extreme example is the space $\beta\mathbb N$, the Stone-Cech compactification of the discrete space $\mathbb N$.

The sequence $x_n = n$ in that space has no convergent subsquence; but (since $\beta\mathbb N$ is compact) it does have a convergent subnet.

So, to answer your question: Yes, there are nets in $\mathbb N$ that are not sequences.