Construct retraction of $conv(X)$ onto $X=[0,1]\times\{0\}\cup\bigcup_{n\in\mathbb{N}}\{1/n\}\times[-1/n,1/n]$

Question

Let $X=[0,1]\times\{0\}\cup\bigcup_{n\in\mathbb{N}}\{1/n\}\times[-1/n,1/n]$ be subset of $\mathbb{R^2}$. And let $Conv(X)$ be a convex hull of $X$. Construct retraction of this convex hull onto $X$.

If retraction must be continuous, is it even true that such retraction exists? I cannot see it.

Answer 1

If you have a box canyon {0,1}×[0,1] $\cup$ [0,1]×{0},
then a continuous retract from the unit square of the
canyon onto the canyon walls can be effected by an ever
deeping U starting from the top [0,1]×{1}.

Adjust this to each of the box canyons of X and simutanously
apply for all those retractions to obtain the beseached retraction.