Two infinite families of equivalences of the continuum hypothesis

Abstract

In this brief note we present two infinite families of equivalences of the Continuum Hypothesis, as follows:$\bullet$ For every fixed $n\ge 2$, the Continuum Hypothesis isequivalent to the following statement: "There is an$n$-dimensional real normed vector space $E$ including a subset$A$ of size ${\mathrm{\aleph }}_1$ such that $E\setminus A$ is not pathconnected''.$\bullet$ For every fixed $T_1$ first-countable topological space$X$ with at least two points, the Continuum Hypothesis isequivalent to the following statement: "There is a point of theTychonoff product $X^{\mathbb{R}}$ with a fundamental system of openneighbourhoods $B$ of size ${\mathrm{\aleph }}_1$''.