On α-compact sets in ideal topological spaces

Abstract

This paper examines the properties of α-continuous functions modulo (J, I) that map a countably α-compact ideal space (X, τ, J) to an ideal space (Y, σ, I), where Y is an α-closed subset of the Cartesian product (X × Y, τ × σ, J × I). It is shown that if (X, τ, J) has a weight of at least ℵ₀, it is the α-continuous image of a closed subspace of the cube D^{ℵ₀}. Additionally, an α-continuous function f: (X, τ, J) → (Y, σ, I), where Y is countably α-compact, can be extended under specific conditions. The concept of α-pseudocompactness is introduced in an ideal topological space (X, τ, J), and it is established that countably α-pseudocompactness is neither finitely multiplicative nor hereditary with respect to α-closed sets. Furthermore, it is proven that an α-continuous function modulo ideals mapping a Tychonoff ideal space to a countably α-pseudocompact space is perfect, and the Tychonoff space itself is countably α-pseudocompact.