To prove the that An open in Banach space if connected if and only if is path-connected $C^{\infty}$ I'd like to prove the following :
$\textbf{Lemma :}$ Let $U$ be open in a Banach space and $D \subset U$ be dense. If $x,y \in U$ are connected through an arch $\gamma \subset U$ it exists a polygonal $\delta \subset U$ which connects $x,y$ whose junction points are in $D$.
I think I was able to find a relatively simple costruction of $\delta$, the problem is that in the junction points the function could be not $C^{2}$.
(The real task was to prove that, given an homotopy $C^{0}$ between two curves $\gamma_{1},\gamma_{2} : [a,b] \longmapsto (X,d)$ without loss of generality I can take a $C^{2}$ homotopy using the Lemma above since, $C^{2}$ function are dense in $C^{0}$)
I think this can be done using the function $e^{-\frac{1}{x^{2}}}$ because it has all null derivatives in $0$, but I don't know how since this argument is new to me.
Any help, hint, reference or other method to the real purpose will be appreciated.