Some continuous decompositions of the space $E^n$

The main result proved is the following. Let $E_f^{(n)}$ ($n>1$) be a continuous decomposition of $E^{(n)}$ into points and zero-dimensional compact sets $\xi_\lambda$. If $P^*=\bigcup\limits_\lambda\xi_\lambda$ is compact and $\mathrm{dim}\,f(P^*)=0$, then the space $f(E^n)$ can be imbedded in $E^{(n+1)}$.