Two infinite families of equivalences of the continuum hypothesis (Q2815273)

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 \(X=\mathbb R\).