On monotonous separately continuous functions

• Main Author: Yaroslav I. Grushka
• Format: Article
• Language: English
• Published: Universitat Politècnica de València 2019-04-01
• Series: Applied General Topology
• Subjects: separately continuous mappings; linearly ordered topological spaces; Young's theorem
• Online Access: https://polipapers.upv.es/index.php/AGT/article/view/9817

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.