Application of selection principles in the study of the properties of function spaces

Abstract

For a Tychonoff space X, we denote by C p (X) the space of all real-valued continuous functions on X with the topology of pointwise convergence.In this paper we prove that:• if every finite power of X is Lindelëof then C p (X) to be strongly sequentially separable iff X is γ-set.• B α (X) -functions Baire class α (1 < α ≤ ω 1 ) on a Tychonoff space X with the pointwise topology -to be sequentially separable iff there exists a Baire isomorphism class α from a space X onto a σ-set.• B α (X) is strongly sequentially separable iff iw(X) = ℵ 0 and X is a Z α -cover γ-set for 0 < α ≤ ω 1 .• there is a consistent example of a set of reals X, such that C p (X) is strongly sequentially separable, but B 1 (X) is not strongly sequentially separable.• B(X) is sequentially separable, but is not strongly sequentially separable for a b-Sierpiński set X.

Projective versions of the properties in the Scheepers Diagram

“…Let X be a set Y with the topology of Sorgenfrey line. Then the space X such that X satisfies S 1 (Ω ω cz , Γ) and iw(X) = ℵ 0 , but X 2 is not Lindelöf (Theorem 6.1 in [15]). Since Γ ω cz ⊂ Ω ω cz , by Proposition 8.4, X projectively S 1 (Ω, Ω).…”

On Ramsey properties, function spaces, and topological games

“…In the papers [2,3,6,16,[22][23][24][25][26]32] various selection principles for a Tychonoff space X were related to the selection principles for C p (X). Likewise, in [10,17,26,30,31] various selection games for X and C p (X) and a bitopological space (C(X), τ k , τ p ) were related.…”

On selective sequential separability of function spaces with the compact-open topology

“…In a series of papers it was demonstrated that γ-covers, Borel covers, k-covers play a key role in function spaces ( [5,8,9,10,12,13,15,17,18,19,20,21,22] and many others). We continue to investigate applications of k-covers in function spaces with the compact-open topology.…”