RETRACTED CHAPTER: On e-Closed Spaces and (θ-e-Continuous Functions

Almost strongly θ -e-continuous functions

“…We introduce and investigate some fundamental properties of almost strongly θ-e-continuous functions defined via e-open sets introduced by Ekici [8] in a topological space. It turns out that almost strong θe-continuity is stronger than θ-e-continuity [11] and weaker than strong θ-e-continuity [19], almost strong θ-semicontinuity [4] and almost strong θ-precontinuity [21]. Moreover, we obtain some results related to separation axioms and graphs properties.…”

“…Separation Axioms Definition 6.1. A space X is called almost e-regular [11] if for any regular closed set F ⊂ X and any point x ∈ X \ F , there exist disjoint e-open sets U and V such that x ∈ U and F ⊂ V . Theorem 6.2.…”

On Weakly e*-θ-open Functions and Their Characterizations

“…A subset A is called e-regular [18] if it is e-open and e-closed. Also, it is noted in [13] that C(X, x), eθO(X, x), eθC(X, x), eR(X, x), RO(X, x), RC(X, x), δO(X, x), δC(X, x), θO(X, x), θC(X, x), SO(X, x), SC(X, x), P O(X, x), P C(X, x), BO(X, x), BC(X, x), eO(X, x), eC(X, x), aO(X, x), aC(X, x)).…”

“…We shall use the well-known accepted language almost in the whole of the proofs of the theorems in this article. The following fundamental properties of e-θ-closure is useful in the sequel: Lemma 2.1 [18]; [13] Let A and B be subsets of a space X. Then the followings are hold: (b) strongly θ-continuous (briefly, st.θ.c.)…”

“…It follows by ( 1) ⇔ (7) that f is weakly eR continuous. ✷ Definition 3.3 [13] A function f : X → Y is called to be contra eθ-continuous (briefly c.eθ.c.…”