TWO INFINITE FAMILIES OF EQUIVALENCES OF THE CONTINUUM HYPOTHESIS
In this brief note we present two infinite families of equivalences of the Continuum Hypothesis, as follows:
• For every fixed n ≥ 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 א1 such that E \ A is not path connected”.
• For every fixed T1 first-countable topological space X with at least two points, the Con- tinuum Hypothesis is equivalent to the following statement: “There is a point of the Tychonoff product XR with a fundamental system of open neighbourhoods B of size א1”.
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2012
