---
abstract: 'In 1959, Pelczynski and Semadeni proved a theorem in which they gave some equivalent conditions for a compact Hausdorff space to be scattered. The purpose of the current note is that to clarify the meaning of the subtle term “conditionally weakly sequentially compact” they used as the basis for the proof of their theorem. Unfortunately, the term now is taken over by a similar but subtle concept that may cause a serious problem.'
address: 'Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada.'
author:
- Fouad Naderi
title: a note on spaces of continuous functions on compact scattered spaces
---

The main result
===============

A locally compact Hausdorff topological space $\varOmega$ is [*scattered*]{} if $\varOmega$ does not contain any non-empty perfect subset (i.e., a closed non-empty subset $\varPi$ of $\varOmega$ such that each point of $\varPi$ is an accumulation point of $\varPi$). Equivalently, any non-empty subset of $\varOmega$ contains at least one isolated point. Some authors use the term [*dispersed*]{} instead of scattered. For more details on scattered spaces see [@Mont], [@Semadeni] and [@Semadeni_book].

\[infinity\] Consider $\mathbb{N}$ with its usual topology and its one point compactification ${\mathbb{N}}^{*}=\mathbb{N} \cup \{\infty\}$. Both $\mathbb{N}$ and ${\mathbb{N}}^{*}$ are scattered. Also, $\{\frac{1}{n}: n\in \mathbb{N} \}\cup \{0\} $ is another compact scattered space which is not discrete. It can be shown that a compact metric space is scattered if and only if it is countable [@Mont p.737]. A scattered space is always totally disconnected. The converse of this is not true as seen by the set $\mathbb{Q}$ of rational numbers.

Consider the following two definitions for the term [*conditionally weakly sequentially compact*]{}.

\[ES\] Let $\varOmega$ be a compact Hausdorff space. We say that $C(\varOmega)$ has [*conditionally weakly sequentially compact property in the sense of*]{} in the sense that for every bounded sequence $(x_n)$ of elements of $C(\varOmega)$ there exists a subsequence $(x_{n_{k}})$ and a member $ x_0 \in C(\varOmega)$ such that for every bounded linear functional $\xi$ on $C(\varOmega)$ the sequence of numbers $\xi(x_{n_{k}})$ converges to $\xi(x_0)$. In other words, if $S$ is a bounded subset of $C(\varOmega)$, then $S$ must be conditionally (=relatively) weakly sequentially compact in the modern language.

\[PS\] Let $\varOmega$ be a compact Hausdorff space. We say that $C(\varOmega)$ has [*conditionally weakly sequentially compact property in the sense of*]{} if for every bounded sequence $(x_n)$ of elements of $C(\varOmega)$ there exists a subsequence $(x_{n_{k}})$ such that for every bounded linear functional $\xi$ on $C(\varOmega)$ the sequence of numbers $\xi(x_{n_{k}})$ is convergent.

The way Definition \[PS\] seems is now clear, but it was after a correspondence with Professor Semadeni that I could write down it this way. In the Definition \[ES\], the space is [**weakly sequentially complete**]{} while in the second one we do not need such a strong condition (not to the point $x_0$). Meanwhile, we use Definition \[ES\] in Eberlin-Smulian theorem to assure weakly compactness of a given set.

\[scatter\] Let $\varOmega$ be a compact Hausdorff space and $C(\varOmega)$ have conditionally weakly sequentially compact property in the sense of . Then $\varOmega$ is finite.

[**Proof.**]{} Suppose to the contrary that $\varOmega$ is infinite. Then, the Banach space of continuous functions $C(\varOmega)$ is infinite dimensional. It is well-known that the weak closure of the unit sphere of an infinite dimensional Banach space is the closed unit ball of the space [@Conway p.128]. Therefore, if $S$ is the unit sphere of $C(\varOmega)$, then ${\overline{S}}^{wk}$ is equal to the unit ball $B$ of $C(\varOmega)$. Since $C(\varOmega)$ has propert , ${\overline{S}}^{wk}=B$ is weakly sequentially compact. According to Eberlin-Smulian Theorem [@Conway p.163], $B$ must be weakly compact. Therefore by [@Conway Theorem 4.2, p.132], $C(\varOmega)$ must be reflexive. By [@Conway p.90], $\varOmega$ must be finite. But, this contradicts our assumption that $\varOmega$ was infinite. Hence, $\varOmega$ can only be a finite set.$\blacksquare$

Suppose $\varOmega$ is a compact Hausdorff space and the main theorem of [@Semadeni p.214] holds. Condition (0) and (9) of the latter theorem assert that $\varOmega$ is scattered if and only if $C(\varOmega)$ has conditionally weakly sequentially compact property in the sense of [**PS**]{}. But, if one wants to classify scatteredness of $\varOmega$ in the sense of [**EP**]{}, (s)he would always end up with a finite set! which is not always the case as Example \[scatter\] indicates.

[**Acknowledgments.**]{} The author would like to thank Professor Z. Semadeni for his careful comments. He would also thanks Professor W. Zelazko who made this corespondents possible.

[1]{}

J. B. Conway, [*A course in functional analysis,*]{} Springer-Verlag, New York, 1990.

V. Montesinos, P. Zizler and V. Zizler, [*An Introduction to Modern Analysis,*]{} Springer International Publishing, Switzerland, 2015.

A. Pelczinsky and Z. Semadeni, [*Spaces of continuous functions III,*]{} Studia. Math. 18 (1959), 211-222.

Z. Semadeni, [*Banach spaces of continuous functions,*]{} Vol. I. Monografie Matematyczne, PWN, Warsaw, 1971.
