Baire property of spaces of continuous functions
We consider the Baire property, pseudocompleteness, and ''convex analogues'' of the Baire property. \(C_ p(X)\) is the space of continuous real functions on the Tikhonov space X in the topology of pointwise convergence. A homogeneous space (in particular, \(C_ p(X))\) is Baire, if it is of the second category. The Baire property and pseudocompleteness of \(C_ p(X)\) were considered by \textit{D. J. Lutzer} and \textit{R. A. McCoy} [Pac. J. Math. 90, 145-168 (1980; Zbl 0481.54017)]. They introduced for any arbitrary topological space X a game \(\Gamma\) (X), in which two players I and II in turn choose finite subsets \(S_ i\), \(i=0,1,..\). \((S_{2i}\) are chosen by player II, \(S_{2i+1}\) by player I), possibly empty, in such a way that \(S_ i\cap S_ j=\emptyset\), if \(i\neq j\). Player I wins if \(\cup^{\infty}_{i=0}S_{2i+1}\) is not a closed discrete subset of X. They proved that if \(C_ p(X)\) is Baire, then player I has no winning strategy in the game \(\Gamma\) (X). The converse does not hold in general. We consider a modification of the game \(\Gamma\) (X), which is the game \(\Gamma_ 1(X)\) in which player I wins if \(\cup^{\infty}_{i=0}S_{2i+1}\) is not strongly discrete (A\(\subseteq X\) is strongly discrete if one can find a discrete system of neighborhoods \(\{\) O(x): \(x\in A\}).\) Main theorem. The following conditions are equivalent: (I) \(C_ p(X)\) is a space of the first category, (II) player I has a winning strategy in the game \(\Gamma_ 1(X)\), (III) there exists a disjoint sequence of nonempty finite sets \(\Delta_ n\subseteq X\), \(n\in N\), such that for any function \(f\in C(X)\sup_{n} \min \{| f(x)|:x\in \Delta_ n\}<\infty\).
scientific article; zbMATH DE number 3969654