Parametric Bing and Krasinkiewicz maps: revisited

Abstract

Abstract. Let M be a complete metric AN R-space such that for any metric compactum K the function space C(K, M ) contains a dense set of Bing (resp., Krasinkiewicz) maps. It is shown that M has the following property: If f : X → Y is a perfect surjection between metric spaces, then C(X, M ) with the source limitation topology contains a dense G δ -subset of maps g such that all restrictions g|f −1 (y), y ∈ Y , are Bing (resp., Krasinkiewicz) maps. We apply the above result to establish some mapping theorems for extensional dimension.

Generalized Cantor manifolds and indecomposable continua

“…Let h : X → I be a continuous function with h(A) = 0 and h(B) = 1. According to [32], there exists a function g : X → I such that |g(x) − h(x)| < 1/4 for all x ∈ X and the restrictions g y = g|f −1 (y) : f −1 (y) → I, y ∈ Y , satisfy the following conditions: the fibers of g y are hereditarily indecomposable, and any continuum K ⊂ f −1 (y) either is contained in a fiber of g y or contains a component of a fiber of g y . Then…”