  It is proved that if X is a compact Hausdorff space of Lebesgue dimension
$\dim(X)$, then the squaring mapping $\alpha_{m} \colon
(C(X)_{\mathrm{sa}})^{m} \to C(X)_{+}$, defined by $\alpha_{m}(f_{1},...,
f_{m}) = \sum_{i=1}^{m} f_{i}^{2}$, is open if and only if $m -1 \ge \dim(X)$.
Hence the Lebesgue dimension of X can be detected from openness of the squaring
maps $\alpha_m$. In the case m=1 it is proved that the map $x \mapsto x^2$,
from the self-adjoint elements of a unital $C^{\ast}$-algebra A into its
positive elements, is open if and only if A is isomorphic to C(X) for some
compact Hausdorff space X with $\dim(X)=0$.
