On monotonous separately continuous functions

• Autores: Yaroslav I. Grushka
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 20, Nº. 1, 2019, págs. 75-79
• Idioma: inglés
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• Resumen

  • Let T = (T, ≤) and T1= (T1 , ≤1) be linearly ordered sets and X be a topological space.  The main result of the paper is the following: If function ƒ(t,x) : T × X → T1 is continuous in each  variable (“t” and  “x”)  separately  and  function ƒx(t)  = ƒ(t,x) is  monotonous  on T for  every x ∈ X,  then ƒ is  continuous  mapping  from T × X to T1,  where T and T1 are  considered  as  topological  spaces  under  the order topology and T × X is considered as topological space under the Tychonoff topology on the Cartesian  product of topological spaces T and X.