Tychonoff expansions by independent families

A method for Tychonoff expansions using independent families is introduced. Using this method we prove that every countable Tychonoff space which admits a partition into infinitely many open-hereditarily irresolvable dense subspaces has a Tychonoff expansion that is ω \omega -resolvable but not strongly extraresolvable. We also show that, under Luzin’s Hypothesis ( 2 ω 1 = 2 ω 2^{\omega _1} = 2^\omega ), there exists an ω \omega -resolvable Tychonoff space of size ω 1 \omega _1 which is not maximally resolvable.