A note on spaces of continuous functions on compact scattered spaces

TL;DR

This paper clarifies the meaning of a subtle term used in a 1959 theorem about compact scattered spaces, addressing potential confusion caused by evolving terminology in the field.

Contribution

It provides a detailed analysis of the term 'conditionally weakly sequentially compact' to prevent misunderstandings in related mathematical proofs.

Findings

Clarification of the term's original meaning

Identification of potential issues with current terminology

Improved understanding of compact scattered spaces

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.