Let a compact Hausdorff space X contain a non-empty perfect subset. If α < β and β is a countable ordinal, then the Banach space Bα (X) of all bounded real-valued functions of Baire class α on X is a proper subspace of the Banach space Bβ (X). In this paper it is shown that: 1. Bα (X) has a representation as C(bα X), where bα X is a compactification of the space P X – the underlying set of X in the Baire topology generated by the Gδ -sets in X. 2. If 1 ≤ α < β ≤ Ω, where Ω is the first...

2000 Mathematics Subject Classification: Primary: 46B03, 46B26. Secondary: 46E15, 54C35.We study the existence of pointwise Kadec renormings for Banach spaces of the form C(K). We show in particular that such a renorming exists when K is any product of compact linearly ordered spaces, extending the result for a single factor due to Haydon, Jayne, Namioka and Rogers. We show that if C(K1) has a pointwise Kadec renorming and K2 belongs to the class of spaces obtained by closing the class of compact...

We prove that the set of all Krasinkiewicz maps from a compact metric space to a polyhedron (or a 1-dimensional locally connected continuum, or an n-dimensional Menger manifold, n ≥ 1) is a dense $G_{\delta }$-subset of the space of all maps. We also investigate the existence of surjective Krasinkiewicz maps from continua to polyhedra.

This paper completes and improves results of [10]. Let $(X,d_{{}_X})$, $(Y,d_{{}_Y})$ be two metric spaces and $G$ be the space of all $Y$-valued continuous functions whose domain is a closed subset of $X$. If $X$ is a locally compact metric space, then the Kuratowski convergence ${\tau }_{{}_K}$ and the Kuratowski convergence on compacta ${\tau }_{{}_K}^c$ coincide on $G$. Thus if $X$ and $Y$ are boundedly compact metric spaces we have the equivalence of the convergence in the Attouch-Wets topology ${\tau }_{{}_{AW}}$ (generated by the box metric of $d_{{}_X}$ and $d_{{}_Y}$) and ${\tau }_{{}_K}^c$ convergence on $G$,...

We prove that, for every countable ordinal α ≥ 3, there exists countable completely regular spaces Xα and Yα such that the spaces Cp (Xα ) and Cp (Yα ) are borelian of class exactly Mα , but are not homeomorphic.

We give an example of a compact space X whose iterated continuous function spaces $C_p\left(X\right)$, $C_p{C}_p\left(X\right),...$ are Lindelöf, but X is not a Corson compactum. This solves a problem of Gul’ko (Problem 1052 in [11]). We also provide a theorem concerning the Lindelöf property in the function spaces $C_p\left(X\right)$ on compact scattered spaces with the ${\omega }_1$th derived set empty, improving some earlier results of Pol [12] in this direction.

For a Tychonoff space $X$, $C\left(X\right)$ is the lattice-ordered group ($l$-group) of real-valued continuous functions on $X$, and $C^{*}\left(X\right)$ is the sub-$l$-group of bounded functions. A property that $X$ might have is (AP) whenever $G$ is a divisible sub-$l$-group of $C^{*}\left(X\right)$, containing the constant function 1, and separating points from closed sets in $X$, then any function in $C\left(X\right)$ can be approximated uniformly over $X$ by functions which are locally in $G$. The vector lattice version of the Stone-Weierstrass Theorem is more-or-less equivalent...

We present a few results and problems related to spaces of continuous functions with the topology of pointwise convergence and the classes of LΣ(≤ ω)-spaces; in particular, we prove that every Gul’ko compact space of cardinality less or equal to $$𝔠$$ is an LΣ(≤ ω)-space.

For a Tychonoff space $X$, let $\downarrow {\mathrm{C}}_F\left(X\right)$ be the family of hypographs of all continuous maps from $X$ to $[0,1]$ endowed with the Fell topology. It is proved that $X$ has a dense separable metrizable locally compact open subset if $\downarrow {\mathrm{C}}_F\left(X\right)$ is metrizable. Moreover, for a first-countable space $X$, $\downarrow {\mathrm{C}}_F\left(X\right)$ is metrizable if and only if $X$ itself is a locally compact separable metrizable space. There exists a Tychonoff space $X$ such that $\downarrow {\mathrm{C}}_F\left(X\right)$ is metrizable but $X$ is not first-countable.