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Phantom mappings and a shape-theoretic problem concerning products. (English) Zbl 1304.54034

In this paper the author studies a shape theoretic problem concerning products. First he formulates eight problems and establishes some equivalences between them; some of these problems offer a new perspective on the subject and some of them are open problems (hopefully to be solved in the future). The fourth problem is the main focus of the study; the proof that problems 6, 7 and 8 are equivalent to problem 4 is also included.
Problem 4 states: “Does there exist a connected polyhedron \((Z,*)\) (weak topology) and a shape morphism \(H:Z\rightarrow H \times P\) such that \(S[\pi_H]H=S[*]\), \(S[\pi_P]H=S[*]\) and \(H\neq S[*]\)?”. Here \((H,*)\) is a Hawaiian earring, \((P,*)\) is a sequence of 1-spheres and \(H\times P\) is the Cartesian product. The affirmative answer for this problem would imply that the Cartesian product \(H \times P\) is not a product in the shape category of topological spaces.
The main result of the paper establishes the equivalence between shape theoretic problems and a problem involving phantom maps (denoted as problem 8 in this paper).

MSC:

54B10 Product spaces in general topology
54B35 Spectra in general topology
54C56 Shape theory in general topology
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