On inductive dimension modulo a simplicial complex

For a given simplicial complex K, V.V. Fedorchuk has recently introduced the dimension functions K-dim and K-Ind and constructed a first countable and separable continuum X n such that K - dim X n = n < 2 n − 1 ⩽ K - Ind X n ⩽ 2 n for each integer n > 1 , provided the join K ⁎ K is non-contractible. We study a modification K - Ind 0 of K-Ind and develop its theory to a point that enables us to compute the inductive dimensions of a variety of spaces. Let α , β be ordinals of cardinality at most c and n an integer with 1 ⩽ n ⩽ α ⩽ β . We construct, inter alia,(1) countable and separable continua S a with K - dim S α = 1 while K - Ind S α = K - Ind 0 S α = α , countable and separable continua S n , α with K - dim S n , α = n while K - Ind S n , α = K - Ind 0 S n , α = α , countable, hereditarily strongly paracompact continua T α such that K - dim T α = K - Ind T α = 1 while K - Ind 0 T α = α , countable continua T n , α , β with K - dim T n , α , β = n while K - Ind T n , α , β = α , and K - Ind 0 T n , α , β = β . he construction of the spaces S α and T α it suffices to assume that K is non-contractible, while the construction of the spaces S n , α and T n , α , β for n > 1 requires the stronger restriction that the join K ⁎ K is non-contractible.