Originally Posted by

**xianghu21** True/False:

1. Is every countable first countable $\displaystyle T_3$-space metrizable?

2. Is every separable first countable $\displaystyle T_3$ space metrizable?

Attempt

1. countable + first countable $\displaystyle \Rightarrow$ second countable. second countable + $\displaystyle T_3$ $\displaystyle \Leftrightarrow T_4$. So by Urysohn Theorem, since this is second countable + $\displaystyle T_4$, this is metrizable. But this argument doesn't work b/c $\displaystyle T_3 \not \Rightarrow T_4$ in general. So is this false? What is the counterexample?

2. separable + first countable $\displaystyle \Rightarrow$ second countable. Now, the argument is the same as above.

Thanks.