It is proved that for a $T_1$-space $X$ the following statements are equivalent: (1) $X$ is PF-normal, (2) for every simplicial complex $K$, every locally selectionable simplex-valued mapping $\vphi:X \to 2^{|K|_w}$ admits a continuous selection $f:X \to |K|_w$, (3) for every simplicial complex $K$, every lower semicontinuous simplex-valued mapping $\vphi :X \to 2^{|K|_m}$ admits a continuous selection $f:X \to |K|_m$. We also characterize finite-dimensional spaces, pseudofinitistic spaces, strongly countable-dimensional spaces, spaces with strong large transfinite dimension and locally finite-dimensional spaces in terms of simplex-valued mappings.