Chapter 30 - Questions in and out of context

Abstract

This chapter discusses some questions from topology that may seem in and out of context. Some of them are just isolated questions that may not lead anywhere; some are missing links in different developments. The first conjecture discussed in the chapter is: “Let X be a tree-like continuum and let f: X →X be a continuous map. Then there exists an indecomposable continuum W ⊆X such that f(W) ∩ W ≠ ϕ.” Second conjecture discussed was posed by Norman Passmore, in his Ph.D. dissertation in 1976, and is still open. It states: “Let P be a pseudo-arc and let C(P) denote its hyperspace of sub-continua. Let X ⊆C(P) be a pseudo-arc. Then X is a subset of a Whitney level for some Whitney map μ.” Third conjecture discussed was posed by Howard Cook and states: “Let A ⊆ℝn be a compact set with the property that every component of A is either a point or a pseudo-arc. Then there exists a pseudo-arc P with A ⊆P ⊂ℝn.” Various other conjectures and problems are also solved.