On inductive dimension modulo a simplicial complex

摘要

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 < 2n - 1 ≤ K-Ind X_n ≤ 2n for each integer n > 1, provided the join K * K is non-contractible. We study a modification K-Indo 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) first countable and separable continua S_α with K-dimS_α = 1 while K-_ndS_α = K-IndoS_α =a, (2) first countable and separable continua S_(n.α) with K-dimS_(n.α) =n while K-Ind S_(n.α) = K-IndoS_(n.α) =a, (3) first 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, (4) first countable continua T_(n.α.β) with K-dm T_(n.α.β) =n while K-Ind T_(n.α.β) = α, and K-Ind_0T_(n.α.β)=β. For the construction of the spaces T_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.