Metric Space is T5

Theorem
Let $M = \left({X, d}\right)$ be a metric space.

Then $M$ is a $T_5$ space.

Proof
Let $S, T \subseteq X$ such that $S$ and $T$ are separated in $X$.

Then:
 * each point $x \in S$ has an open $\epsilon$-ball $B_{\epsilon_x} \left({x}\right)$ which is disjoint from $T$


 * each point $y \in T$ has an open $\epsilon$-ball $B_{\epsilon_y} \left({y}\right)$ which is disjoint from $S$.

Then:
 * $U_S = \displaystyle \bigcup_{x \mathop \in S} B_{\epsilon_x / 2} \left({x}\right)$


 * $U_T = \displaystyle \bigcup_{y \mathop \in T} B_{\epsilon_y / 2} \left({y}\right)$

are disjoint open neighborhoods of $S$ and $T$ respectively.

Hence the result by the definition of $T_5$ space.