Spaces determined by selections |
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Authors: | Michael Hru&scaron á k,Ivá n Martí nez-Ruiz |
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Affiliation: | Instituto de Matemáticas, UNAM, Apartado Postal 61-3, Xangari, 58089, Morelia, Michoacán, Mexico |
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Abstract: | A function is a called a weak selection if ψ({x,y})∈{x,y} for every x,y∈X. To each weak selection ψ, one associates a topology τψ, generated by the sets and . Answering a question of S. García-Ferreira and A.H. Tomita [S. García-Ferreira, A.H. Tomita, A non-normal topology generated by a two-point selection, Topology Appl. 155 (10) (2008) 1105-1110], we show that (X,τψ) is completely regular for every weak selection ψ. We further investigate to what extent the existence of a continuous weak selection on a topological space determines the topology of X. In particular, we answer two questions of V. Gutev and T. Nogura [V. Gutev, T. Nogura, Selection problems for hyperspaces, in: E. Pearl (Ed.), Open Problems in Topology 2, Elsevier B.V., 2007, pp. 161-170]. |
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Keywords: | primary, 54C65 secondary, 54B20 |
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