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.**