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 $\aleph_1$''.

Keywords: Continuum Hypothesis; path connected subsets; normed spaces; $T_1$ spaces; product topology; function spaces.

MSC: 03E50, 54A35, 54B10, 54C30

Pages:  109$-$112     

Volume  66 ,  Issue  1 ,  2014