Prove that the sets are open. I'm having some trouble with the following questions:
Let $S$ be any set and $\epsilon$ > 0. Define $T$ = {$t$ $\in$ $\mathbb{R}$ : |$t - s$| < $\epsilon$ for some $s \in S$}. Prove that T is open. 
Now, again let $S$ be any set and define $V$ = {$t$ $\in$ $\mathbb{R}$ : |$t - s$| $\le$ 1 for some $s \in S$}. Is $V$ necessarily closed?
Thanks you for any help you can provide!

Hint for 1): $T$ is the union of the open balls $B_s=\{ x :|x-s|<\epsilon\}$, $s\in S$.
Hint for 2): Consider $S=(0,1)$ in $\Bbb R$.  Is $2\in V$? Is $(0,2)\subseteq V$?
