Chain Conditions in Cₚ(X)

We present new results regarding calibers in function spaces Cₚ(X). We calculate the calibers of Cₚ(X) when X is an interval of ordinals and when X is the one-point λ-Lindel\"of extension of a discrete space of cardinality ≥ λ. We give sufficient conditions to characterize the calibers of Cₚ(X) when X is a topological sum, and we calculate the calibers of Cₚ(X) when X = ∏ξ < λX_ξ is a product of non-trivial Tychonoff spaces with i-weight less or equal to λ. The main theorem is: If κ and λ are cardinals with ω ≤ λ≤ κ, then the set of calibers of the space of the real-valued continuous functions defined on the one-point λ-Lindel\"of extension of the discrete space of cardinality κ with its pointwise convergence topology, Cₚ(L(λ, κ)), is \μ∈ CN : cf(μ)>ω \ \ μ∈ (λ ,κ] \ \ cf(μ)∈ (λ,κ]\.