On δ-I-open sets and decomposition of α-I-continuity

A note on δ-I-sets

“…δ-set and δ-I-set are independent concepts [1, Remark 2.2]. The following Theorem 2.1 shows that every δ-set is a δ-I-set, if the ideal is codense and Example 2.3 of [1] says that the reverse inclusion is not true, even if the ideal is codense. Theorem 2.2 below shows that for completely codense ideals, δ-I-sets coincide with δ-sets and so Proposition 2.6 of [1] becomes a corollary to Theorem 2.2.…”

“…In [1,Proposition 2.4], it is established that if A is a δ-I-set in an ideal space (X, τ, I), then A = B ∪ C, where B ∈ αIO(X), int cl (C) = ∅ and B ∩ C = ∅. The following theorem gives a necessary condition for a set to be a δ-I-set.…”

Onα-Layer Ideal Topologies

“…After M. E. Abd El-Monsef introduced the concept of I-open set, many researchers were devoted to research on local semitopology. There are many local semiopen sets, such as β-I-open set, strong β-I-open set, and δ-I-open set [3][4][5]. All of these local semiopen sets are given by comparing interior and closure operators in the initial topology and its ideal derived topology.…”

Decompositions of Some Forms of Continuity

“…Quite recently, Acikgoz and Yuksel [3] have introduced I -R closed sets and obtained a decomposition of continuity. Acikgoz et al [1], [2] investigated some properties of I -open sets and obtained decompositions of I -continuity and semi -I -continuity.…”