Decompositions of $\star$-continuity and $\mathcal{A}^{\star}$-$I_\omega$-continuity

Abstract

The aim of this paper is to introduce and study the notions of $\mathcal{A}^\star$-$I_\omega$-sets, $I_\omega$-$\mathcal{C}$-sets, $\eta$-$I_\omega$-sets, $\mathcal{A}^{\star \star}$-$I_\omega$-sets, $\eta^\star$-$I_\omega$-sets, $I_\omega$-$\mathcal{C}^\star$-sets, $\mathcal{C}^{\star \star}$-$I_\omega$-sets and $\mathcal{C}^\star$-$I_\omega$-sets in ideal topological spaces. Properties of such classes of sets are investigated. Moreover, decompositions of $\star$-continuous functions and decompositions of $\mathcal{A}^\star$-$I_\omega$-continuous functions in ideal topological spaces are established.