On the connection between games on a space and on its \(C_p\)-space

Abstract

We present a negative answer to Tkachuk's question if there is a Tychonoff space X such that \(C_p(X)\) has the Fréchet–Urysohn property and Player II has a winning strategy in \(CD(C_p(X))\), as well as a positive one to Clontz and Holshouser’s question whether there is a point-picking game on \(C_p(X)\) which characterizes when X is not a Rothberger space. In reality, we prove a stronger statement: that there is a game on \(C_p(X)\) which is equivalent to the Rothberger game on X. Moreover, we prove the equivalence between pairs of games, one of which played on \(C_p(X)\) and one on X, by providing a two-way “translation” of strategies.