Exactly n-resolvable topological expansions

For a cardinal κ > 1 , a space X = ( X , T ) is κ-resolvable if X admits κ-many pairwise disjoint T -dense subsets; ( X , T ) is exactly κ-resolvable if it is κ-resolvable but not κ + -resolvable. esent paper complements and supplements the authorsʼ earlier work, which showed for suitably restricted spaces ( X , T ) and cardinals κ ⩾ λ ⩾ ω that ( X , T ) , if κ-resolvable, admits an expansion U ⊇ T , with ( X , U ) Tychonoff if ( X , T ) is Tychonoff, such that ( X , U ) is μ-resolvable for all μ < λ but is not λ-resolvable (cf. Comfort and Hu, 2010 [11, Theorem 3.3]). Here the “finite case” is addressed. The authors show in ZFC for 1 < n < ω : (a) every n-resolvable space ( X , T ) admits an exactly n-resolvable expansion U ⊇ T ; (b) in some cases, even with ( X , T ) Tychonoff, no choice of U is available such that ( X , U ) is regular (nor even quasi-regular); (c) if regular and n-resolvable, ( X , T ) admits an exactly n-resolvable regular expansion U if and only if either ( X , T ) is itself exactly n-resolvable or ( X , T ) has a subspace which is either n-resolvable and nowhere dense or is ( 2 n ) -resolvable. In particular, every ω-resolvable regular space admits an exactly n-resolvable regular expansion. Further, for many familiar topological properties P (e.g., Tychonoff; has a clopen basis), one may choose U so that ( X , U ) ∈ P if ( X , T ) ∈ P .