  Let $X$ and $Y$ be metrizable spaces and suppose that there exists a
uniformly continuous surjection $T: C_{p}(X) \to C_{p}(Y)$ (resp., $T:
C_{p}^*(X) \to C_{p}^*(Y)$), where $C_{p}(X)$ (resp., $C_{p}^*(X)$) denotes the
space of all real-valued continuous (resp., continuous and bounded) functions
on $X$ endowed with the pointwise convergence topology.
  We show that if additionally $T$ is an inversely bounded mapping and $X$ has
some dimensional-like property $\mathcal P$, then so does $Y$. For example,
this is true if $\mathcal P$ is one of the following properties:
zero-dimensionality, countable-dimensionality or strong
countable-dimensionality.
  Also, we consider other properties $\mathcal P$: of being a scattered, or a
strongly $\sigma$-scattered space, or being a $\Delta_1$-space (see [17]). Our
results strengthen and extend several results from [6], [13], [17].
