Realcompactness and spaces of vector-valued functions

Realcompactness and Banach-Stone theorems

For realcompact spaces X and Y we give a complete description of the linear biseparating maps between spaces of vector-valued continuous functions on X and Y, where special attention is paid to spaces of vector-valued bounded continuous functions. These results are applied to describe the linear isometries between spaces of vector-valued bounded continuous and uniformly continuous functions.

The Samuel realcompactification of a metric space

In this paper we introduce a realcompactification for any metric space (X, d), defined by means of the family of all its real-valued uniformly continuous functions. We call it the Samuel realcompactification, according to the well known Samuel compactification associated to the family of all the bounded real-valued uniformly continuous functions. Among many other things, we study the corresponding problem of the Samuel realcompactness for metric spaces. At this respect, we prove that a result of Katětov-Shirota type occurs in this context, where the completeness property is replaced by Bourbak…

Recent progress in Rings and Subrings of Real Valued Measurable Functions

Two separated realcompact measurable spaces $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ are shown to be isomorphic if and only if the rings $\mathcal{M}(X,\mathcal{A})$ and $\mathcal{M}(Y,\mathcal{B})$ of all real valued measurable functions over these two spaces are isomorphic. It is furthermore shown that any such ring $\mathcal{M}(X,\mathcal{A})$, even without the realcompactness hypothesis on $X$, can be embedded monomorphically in a ring of the form $C(K)$, where $K$ is a zero dimensional Hausdorff topological space. It is also shown that given a measure $μ$ on $(X,\mathcal{A})$, the $m_μ$-to…

Baire 一函数空间的 Lindelöf 型性质

Lindelöf-type properties of spaces of Baire-one functions

本文研究逐点收敛拓扑下 Baire 一函数空间的 Lindelöf 型性质，探讨其正规性、外延性与实紧性，证明遗传 Baire 及 Polish 可分可度量化空间和 Luzin 集上实紧性成立，回答 Teng 的两个问题并提出开放问题。