MATEMATIČKI VESNIK
МАТЕМАТИЧКИ ВЕСНИК



MATEMATIČKI VESNIK
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 $\aleph_1$''.

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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