Does every net have a countable subnet?

Question

I was wondering whether a net always has a countable subnet? Since there is a criterion for continuity using nets, and in some spaces we can check only for sequences. It would seem to me that a convergent net to a point with a countable local base we can find, but is this even true?

Answer 1

No. The net on $\omega_1$ into $\omega_2,$ sending each
countable ordinal to itself is a counter example.

If a net converges to a point p with countable
local base, then yes, there is a countable subnet.

Let { $U_1, U_2,$ ... } be the countable base and use
the decending local base { $U_1, U_1 \cap U_2,$ ... }
to construct a countable subnet.