Two infinite families of equivalences of the continuum hypothesis

Samuel G. da Silva

Abstract

In this brief note we present two infinite families of equivalences of the Continuum Hypothesis, as follows:
$\bullet$ For every fixed $n \geq 2$, 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''.
$\bullet$ For every fixed $T_1$ 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 $\aleph_1$''.