The topologization of the space of separately continuous functions

Abstract

Here we introduce locally convex topology $\mathcal{T}$ of the layer uniform convergence on the space $ S = CC [0,1] ^ 2 $ of all separately continuous functions $ f: [0,1] ^ 2 \rightarrow \mathbb{R}$, we prove that the space $(S, \mathcal{T}) $ is complete and it is not metrizable one, the space $ P $ of all polynomials of two variables on $ [0,1] ^ 2 $ is everywhere dense in $ S $, and so, $ S $ is separable.