True/False:

1. Is every countable first countable -space metrizable?

2. Is every separable first countable space metrizable?

Attempt

1. countable + first countable second countable. second countable + . So by Urysohn Theorem, since this is second countable + , this is metrizable. But this argument doesn't work b/c in general. So is this false? What is the counterexample?

2. separable + first countable second countable. Now, the argument is the same as above.

Thanks.