Countable-dimensional universal sets

Abstract

The main results of this paper are a construction of a countable union of zero dimensional sets in the Hilbert cube whose complement does not contain any subset of finite dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than-or-slanted-equals 1"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>⩾<!-- ⩾ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n \geqslant 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (Theorem 2.1, Corollary 2.3) and a construction of universal sets for the transfinite extension of the Menger-Urysohn inductive dimension (Theorem 2.2, Corollary 2.4).