Two infinite families of equivalences of the continuum hypothesis
type: Item
instance of: scholarly article
zbMATH Open document ID: 1349.03058
publication date: 2016-06-27T00:00:00Z
Mathematics Subject Classification ID: 03E50
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 54B10
zbMATH DE Number: 6598887
zbMATH Keywords: path connected subset
zbMATH Keywords: normed space
zbMATH Keywords: product topology
author: Samuel Gomes da Silva
MaRDI profile type: Publication
title: Two infinite families of equivalences of the continuum hypothesis
published in: Matematički Vesnik
review text: The author generalizes two well-known equivalents of the continuum hypothesis. First, for every \(n\geq 2\), the continuum hypothesis is equivalent to the following statement: ``There is an \(n\)-dimensional real normed vector space \(E\) including a subset \(A\) of size \(\aleph_1\) such that \(E\setminus A\) is not path connected''; this is a generalization of the statement for \(n=2\). Second, for every \(T_1\) first-countable topological space \(X\) with at least two points, the continuum hypothesis is equivalent to the following statement: ``There is a point of the Tychonoff product \(X^{\mathbb R}\) with a fundamental system of open neighbourhoods \(B\) of size \(\aleph_1\)''; this generalizes the known result for \(X=\mathbb R\).
description: scientific article; zbMATH DE number 6598887