On uniformly continuous surjections between $C_p$-spaces over metrizable spaces

Abstract

Let $X$ be metrizable, $Y$ be perfectly normal and suppose that there exists a uniformly continuous surjection $T:C_p(X)\to C_p(Y)$ (resp., $T:C_p^{\ast }(X)\to C_p^{\ast }(Y)$), where $C_p(X)$ (resp., $C_p^{\ast }(X)$) 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 $\ddot{I}ƒ$-scattered space, or being a $\hat{I}{”}_1$-space (see [17]). Our results strengthen and extend several results from [6], [13], [17].

11 pages