On α-I-continuous and α-I-open functions

Some new sets and decompositions of A I-R-continuity, α-I-continuity, continuity via idealization

“…Let (X, τ ) be a topological space and I an ideal of subsets of X. An ideal I is defined as a nonempty collection of subsets of X satisfying the following two conditions: (1) if A ∈ I and B ⊂ A, then B ∈ I; (2) if A ∈ I and B ∈ I, then A ∪ B ∈ I. An ideal topological space (X, τ ) is a topological space with an ideal I on X and is denoted as (X, τ, I).…”

“…This follows from the fact that if (X, τ, I) is an I-submaximal and P -I-disconnected space, then τ = αIO (X) = PIO (X) = SIO (X) (Acıkgöz et al [2]). Proposition 3.6 (Acıkgöz et al [1]). Let (X, τ, I) be an ideal topological space.…”

“…Acıkgöz et al [1] obtained a decomposition of α-I-continuity. Dontchev [3] has introduced the notion of pre-I-open sets and pre-I-continuity and Keskin et al [7] introduced the concept of weakly I-locally-closed set.…”

Subsets of ideal topological spaces

“…A subset A of an ideal space (X, τ, I) is said to be α-I-open [6] if [1]. It follows that τ and αIO(X, τ ) are α-equivalent [17] and have the same family of regular open sets, preopen sets [9], semipreopen sets [2], dense sets [20] and nowhere dense sets [17].…”

“…In 1990, Janković and Hamlett [10] further studied ideal topological spaces and their applications to various fields. After their study, it was further investigated by many authors in [1], [4], [5], [6], [7], [11], [12] and [13]. In this note, we discuss the properties of α-I-open sets, t-I-sets, strong β-I-open sets, S βI -sets and S-I-sets in ideal topological spaces.…”