About Baire property of space of separately continuous functions

Abstract

We propose three methods for proving that locally convex
space $S = CC [0,1] ^ 2$ of separately continuous functions $f:
[0,1] ^2 \rightarrow \mathbb {R}$ on the square $[ 0,1] ^ 2$ with
topology of layer-wise uniform convergence is the set of the first
category, therefore, is not Baire space. The methods of
$\varepsilon$-nets, function of calculation or topological games
were used in this research. These approaches were generalized on
spaces $CC (X \times Y)$ of separately continuous functions $f: X
\times Y \rightarrow \mathbb {R} $, which were topologized
respectively.