Some remarks on a game of Telgلrsky

In [3] R. Telgársky (1975) asked: does the first player have a winning strategy in the game G ( F , X × X ) if the first player has a winning strategy in the game G ( F , X ) ? I give a positive answer to this question and prove that this result is also true for spaces where the first player has a winning strategy in game G ( K , X ) where K = 1 , F, C, for σ C if X is P-space and for DC if X is collectionwise-normal space. The last result is related to the Telgárskyʹs (1983) conjecture discussed in [1]. These results are not true for infinite product of spaces.