The sequential defect of the cross topology is \(\omega _ 1\)
type: Item
instance of: scholarly article
title: The sequential defect of the cross topology is \(\omega _ 1\)
zbMATH Open document ID: 0569.54028
DOI: 10.1016/0166-8641(85)90088-4
author: Gabriele H. Greco
published in: Topology and its Applications
publication date: 1985-01-01T00:00:00Z
review text: With the aid of inductive limits, \textit{A. V. Arkhangel'skij} and \textit{S. P. Franklin} [Mich. Math. J. 15, 313-320 (1968; Zbl 0167.511)] constructed a countable, sequential, homogeneous Hausdorff space in which the topological closure is obtained by precisely \(\omega_ 1\) iterations of the sequential closure. This paper shows that the rational plane \(Q^ 2\) with the cross topology has the properties stated above. Here the cross topology is the topology in which a set A is open if and only if every ''horizontal'' or ''vertical'' sequence, converging in the usual way to a point in A, is eventually in A.
Mathematics Subject Classification ID: 54D55
Mathematics Subject Classification ID: 54A05
zbMATH DE Number: 3909311
zbMATH Keywords: pretopologies
zbMATH Keywords: topological closure
zbMATH Keywords: sequential closure
zbMATH Keywords: rational plane
zbMATH Keywords: cross topology
reviewed by: Y. Tanaka
MaRDI profile type: Publication
full work available at URL: 0166-8641(85)90088-4
OpenAlex ID: W2028649402
cites work: Ordinal invariants for topological spaces
cites work: Q5788019
cites work: Topologically maximal pretopologies
cites work: Q5341489
cites work: On convergence spaces and their sequential envelopes
description: scientific article; zbMATH DE number 3909311