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.
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Last modified January 5, 2002.