On monotonous separately continuous functions

Yaroslav I. Grushka

doi:10.4995/agt.2019.9817

Authors

Yaroslav I. Grushka Institute of Mathematics NAS of Ukraine

DOI:

https://doi.org/10.4995/agt.2019.9817

Keywords:

separately continuous mappings, linearly ordered topological spaces, Young's theorem

Abstract

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 â†’ T₁is continuous in each variable (“t” and “x”) separately and function ƒ_x(t) = ƒ(t,x) is monotonous on T for every x âˆˆ 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.

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Author Biography

Yaroslav I. Grushka, Institute of Mathematics NAS of Ukraine

Department of Nonlinear analysis

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