Open sets in infinite dimensional spaces

Question

Let $C$ be an closed subset of the Banach space $X$. I am wondering whether the following statements are equivalent for any Banach space:

1. $O$ is an open set containing $C$.

2. For every $x\in \partial C$ and every direction $y$, there exists $\epsilon >0$ such that $x+\epsilon y$ is in $O$.

3. For every sequence $x_n$ converging to $x\in \partial C$, there is an $N$ such that $x_n\in O$, $n\ge N$.

Answer 1

3 does not imply 1.
Let X = R, C = {1}, O = [0,2].
O is a nhood of C could get you better results.