Weakly infinite dimensional subsets of R^N

TL;DR

This paper explores the relationship between first category sets and countable dimensional subsets in infinite-dimensional Euclidean spaces, revealing that their algebraic sum with certain special sets remains of first category under the Continuum Hypothesis.

Contribution

It establishes a duality under the Continuum Hypothesis between first category and countable dimensional sets in infinite-dimensional spaces, and examines the algebraic sum properties of special dimension-theoretic sets.

Findings

The algebraic sum of Hurewicz sets with compactly countable dimensional sets is of first category.

Under the Continuum Hypothesis, a duality exists between first category and countable dimensional sets in 1^N.

Certain dimension-theoretic sets maintain first category status when summed with specific subsets.

Abstract

The Continuum Hypothesis implies an Erd\"os-Sierpi\'nski like duality between the ideal of first category subsets of $\reals^{\naturals}$, and the ideal of countable dimensional subsets of $\reals^{\naturals}$. The algebraic sum of a Hurewicz subset - a dimension theoretic analogue of Sierpinski sets and Lusin sets - of $\reals^{\naturals}$ with any compactly countable dimensional subset of $\reals^{\naturals}$ has first category.