Let $E$ be a Banach space, $X_1, X_2, \ldots $ be a numerable collection of finite dimensional subspaces $X_1\subset X_2$ with dimension tending to infinity, denote by $S^n$ the unit sphere in $X_n$ . Denote by $X$ the union of the $S^n$ and let Y be the completion of the union $\cup X_n$ in the ambient norm in E.
Is there a (weak?) retraction, homotopy equivalence or domination
between X and the unitary sphere in $Y$? Notice that $Y$ is not a CW complex.