Martin's Axiom and a regular topological space with uncountable net weight whose countable product is hereditarily separable and hereditarily Lindelof

by

Krzysztof Ciesielski

J. Symbolic Logic 52(2) (1987), 396-399

In the paper it is proved that if set theory ZFC is consistent then so is the following

ZFC + Martin's Axiom + negation of the Continuum Hypothesis +
there exists a 0-dimensional Hausrorff topological space X such that
X has net weight nw(X) equal to continuum, but
nw(Y)=\omega for every subspace Y of X of cardinality less than continuum.

In particular, the countable product X\omega of X is hereditarily separable and hereditarily Lindelof, while X does not have countable net weight. This solves a problem of Arhangel'skii.