Baire property of spaces of continuous functions

Weak pseudocompactness on spaces of continuous functions

“…The complete characterization of spaces X for which C p (X) is a Baire space was given simultaneously by E. van Douwen in a private communication (see [14]) and E.G. Pytkeev [18]. The characterization of the Baire property for spaces of the form C p (X, Y ) is a pending task.…”

Two properties of Cp(X) weaker than the Fréchet Urysohn property

“…Recall that a family {A α } α∈κ of subsets of a space X is said to be strongly discrete if for every α ∈ κ, there exists an open set U α of X such that A α ⊂ U α and {U α } α∈κ is discrete [14, p. 377]. The theorem below is due independently to van Douwen [4], Pytkeev [18] and Tkachuk [23]. 18,23]).…”

“…The theorem below is due independently to van Douwen [4], Pytkeev [18] and Tkachuk [23]. 18,23]). For a space X, the following are equivalent:…”

The Cantor tree, the 𝛾-property, and Baire function spaces

“…Let Y = co U {q} , where co is the space of all natural numbers and q £ fico, and Y has the topology inherited from Bco, the Stone-Cech compactification of co. It was shown in [10,13] that CP(Y) is a Baire space but not weakly a-favorable. It is true that for any space Z , if CP(Z) is Baire, then Ck(Z) = CP(Z).…”

“…Specifically, we consider function spaces on the Cantor tree. All spaces under consideration are Tychonoff, and our notation and terminology, In [ 13] the Baire property of CP(Y) is characterized in terms of some topological property of the domain space Y. Because this property of Y is preserved under free unions, it follows that the Baire property of CP(Y) is productive, i-e-> ELes Cp(Xa) is a Baire space if Cp(Xa) is a Baire space for each a in S. In this paper, we will show that it is consistent with the usual (ZFC) axioms of set theory that there are Baire function spaces Ck(Y) and Ck(X) such that Ck(X) x Ck(Y) is not Baire.…”