α-CONTINUOUS FUNCTIONS AND COUNTABLY Α-COMPACT SETS IN IDEAL TOPOLOGICAL SPACES

Abstract

In this research work, we introduce the concept of countably α-compact spaces in an ideal topological space (X,τ,I) and study further properties of α-continuous functions. We prove that α-continuous function mapping a countably α-compact ideal topological space (X,τ,I) to the ideal space (Y,σ,J) is an α-closed subset of the Cartesian product (X×Y,τ×σ,I×J). We showed that the countably α-compact ideal space (X,τ,I) with weight ∣X∣≥ ℵ₀ is the α-continuous image of a closed subspace of the cube D^({ℵ₀}) and illustrate that the α-continuous function f:(X,τ,I)→(Y,σ,J) where Y is countably α-compact can be extended over its domain under some constraints. Moreover, α-pseudocompact is defined in an ideal topological space (X,τ,I) and we proved that countably α-pseudocompactness is not hereditary with respect to α-closed sets and we showed that countab α-pseudocompactness is not finitely multiplicative. We find that if the ideal topological space (X,τ,I) is Tychonoff, (Y,σ,J) is countably α-pseudocompact, and f:(X,τ,I)→(Y,σ,J) is α-continuous perfect function modulo (I,J), X is countably α-psuedocompact.