Tychonoff expansions with prescribed resolvability properties

The recent literature offers examples, specific and hand-crafted, of Tychonoff spaces (in ZFC) which respond negatively to these questions, due respectively to Ceder and Pearson (1967) [3] and to Comfort and García-Ferreira (2001) [5]: (1) Is every ω-resolvable space maximally resolvable? (2) Is every maximally resolvable space extraresolvable? Now using the method of KID expansion, the authors show that every suitably restricted Tychonoff topological space ( X , T ) admits a larger Tychonoff topology (that is, an “expansion”) witnessing such failure. Specifically the authors show in ZFC that if ( X , T ) is a maximally resolvable Tychonoff space with S ( X , T ) ⩽ Δ ( X , T ) = κ , then ( X , T ) has Tychonoff expansions U = U i ( 1 ⩽ i ⩽ 5 ), with Δ ( X , U i ) = Δ ( X , T ) and S ( X , U i ) ⩽ Δ ( X , U i ) , such that ( X , U i ) is: ( i = 1 ) ω-resolvable but not maximally resolvable; ( i = 2 ) [if κ ′ is regular, with S ( X , T ) ⩽ κ ′ ⩽ κ ] τ-resolvable for all τ < κ ′ , but not κ ′ -resolvable; ( i = 3 ) maximally resolvable, but not extraresolvable; ( i = 4 ) extraresolvable, but not maximally resolvable; ( i = 5 ) maximally resolvable and extraresolvable, but not strongly extraresolvable.