Given set U is first countable or not?

Question

In $\mathbb{R}$ with usual topology ,the set $U =\{ x \in \mathbb{R} : -1\le x \le 1 , ,x \neq 0\}$ is

Choose the correct statement

$a)$ Neither hausdorff nor First counatble

$b)$ Hausdorff

$c)$ First countable

$d)$both hausdorff and first countable

My attempt :set $U$ can be written as $[-1,0)$ and $(0,1]$ which are two disjoint set, From this i can concnclude that $U$ is hausdorff

Im confusing that it is First countable or not ?

Any hints/solution will be appreciated

thanks u

Answer 1

The usual topology is induced by a metric and every metric space is first-countable.

Answer 2

$U$ can be written as $[-1,0)$ and $(0,1]$ which are two disjoint set, From this i can concnclude that $U$ is hausdorff

The fact that $U$ can be written as a union of two disjoint sets has nothing to do with the set being Hausforff or not.

For the first countable property, google is your friend.

Answer 3

$\mathbb{R}$ with usual topology is also a metric space. So $\mathbb{R}$ is first countable. Hence any subspace is also first countable.