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Disjointness of stars in a simplicial complex in $\ell_2$

Definitions Let's consider a full $n$-dimensional simplicial complex $C$ in $\ell_2(X)$. By that I mean a set of all functions $f:X \to[0,1]$ such that $\sum_{x\in X}f(x)=1$ and there are at most ...
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Dimension of a subset

For a closed subset $Y$ of a space $X$ we have the following inequlity of topological (covering) dimensions: $$\dim{Y} \leq \dim{X}$$ (assuming at least one of those is finite). I have two questions ...