The Scarborough-Stone problem for Hausdorff spaces
type: Item
instance of: scholarly article
title: The Scarborough-Stone problem for Hausdorff spaces
zbMATH Open document ID: 0758.54010
DOI: 10.1016/0166-8641(92)90103-7
author: Peter Nyikos
author: Jerry E. Vaughan
published in: Topology and its Applications
publication date: 1993-02-18T00:00:00Z
review text: \textit{C. T. Scarborough} and \textit{A. H. Stone} [Trans. Am. Math. Soc. 124, 131-147 (1966; Zbl 0151.300)] proved that every product of no more than \(\aleph_ 1\) sequentially compact spaces (no separation axioms assumed) is countably compact. They asked if every product of sequentially compact spaces is countably compact. This is now known as the Scarborough-Stone problem. Previous work has answered the question negatively for the class of completely regular, Hausdorff spaces in many models of set theory. The authors answer the question negatively in ZFC for the class of Hausdorff spaces. They provide two different methods for constructing families of sequentially compact spaces whose product is not countably compact. One of these constructions is a modification of the Ostaszewski technique; the other is an iteration of the \(\Psi\)-space construction.
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54D55
Mathematics Subject Classification ID: 54D80
Mathematics Subject Classification ID: 54D10
zbMATH DE Number: 123584
zbMATH Keywords: countably compact space
zbMATH Keywords: weak base
zbMATH Keywords: Scarborough-Stone problem
zbMATH Keywords: Hausdorff spaces
zbMATH Keywords: sequentially compact spaces
reviewed by: Douglas D. Mooney
MaRDI profile type: Publication
full work available at URL: 0166-8641(92)90103-7
OpenAlex ID: W2062049674
cites work: MAPPINGS AND SPACES
cites work: Q5186348
cites work: Spaces in which sequences suffice
cites work: Q3270120
cites work: On countably compact, locally countable spaces
cites work: Metrizability and the Frechet-Urysohn Property in Topological Groups
cites work: Sequentially Compact, Franklin-Rajagopalan Spaces
cites work: Q4098359
cites work: Products of Sequentially Compact Spaces and The V-Process
cites work: Products of Nearly Compact Spaces
cites work: Q4160059
cites work: Products of Perfectly Normal, Sequentially Compact Spaces
description: scientific article; zbMATH DE number 123584