Openness of induced projections

Abstract

Abstract. For continua X and Y it is shown that if the projection f : X × Y → X has its induced mapping C(f ) open, then X is C * -smooth. As a corollary, a characterization of dendrites in these terms is obtained.All spaces considered in this paper are assumed to be metric. A mapping means a continuous function. To exclude some trivial statements we assume that all considered mappings are not constant. A continuum means a compact connected space. Given a continuum X with a metric d, we let 2 X denote the hyperspace of all nonempty closed subsets of X equipped with the Hausdorff metric H defined by

Induced open projections and C*-smoothness

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We show that there exists a C * -smooth continuum X such that for every continuum Y the induced map C(f ) is not open, where f : X × Y → X is the projection. This answers a question of Charatonik et al (2000).

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“…In [2] some results on the openness of C(π Y X ) were obtained. In [3,Theorem 4], it was proved that if there exists a continuum Y such that C(π Y X ) is open, then X is C * -smooth, and it was asked if the converse holds [3,Problem 6]. In this paper we answer this question in the negative.…”

“…The example. In Problem 6 of [3] it was asked if each C * -smooth continuum X has the open projection property. In the following example, we give a negative answer to this question.…”

“…3 and (π(ψ(t)), 0) ∈ T for each t ∈ [0, ∞). For each m ∈ N, {(m + 1)e 1 , (m + 1)e 2 , (m + 1)e 3 } ⊂ Im σ βm = ξ m ([m − 1, m]).…”