Continuity Via ΛI-Open Sets

Abstract

The complex [Cu(en) 2 (H 2 O)](sy) 2 (en)(H 2 O) 2 has been synthesized and characterized by its electronic and vibrational spectra. The molecular structure of the complex has been determined by X-ray diffraction methods. The complex crystallizes in the orthorhombic space group Pnma with unit-cell parameters a = 10.7236 (5), b = 20.4660(10), c = 14.4523(11)Å and Z = 4. In the cation, the Cu(II) ion has a distorted square pyramidal coordination with two bidendate (en) ligands forming the basal plane and a H 2 O molecule in the apical position. The complex cations and syringate anions constitute chains along the b axis in -A-B-A-fashion. The members of the chains are linked by through N-H···O hydrogen bonds. The (en) molecules are responsible for connecting adjacent layers.

Contra-continuous Functions Defined Through Λ1-closed Sets

“…The complement of a Λ I -closed set is called Λ I -open set. In [10] Next, we present the definitions and characterizations of Λ I -continuous, quasi-Λ I -continuous and Λ Iirresolute functions given in [11]. Definition 2.3.…”

“…In this section we study the behavior of some topological notions under direct or inverse images of the new variants of contra-continuity introduced in the Section 3. Before continuing our study, we must remember the following definitions introduced in [11]. An ideal topological space (X, τ, I) is said to be Λ I -connected (resp.…”

“…We say that an ideal topological space (X, τ, I) is strongly S-Λ I -closed, if each cover of X by Λ I -closed sets has a finite subcover. The notions of Λ I -compact space and τ ⋆ -compact space were introduced in [11]. We say that an ideal topological space (X, τ, I) is Λ I -compact (resp.…”

“…In 1996, Dontchev [2] introduced the notion of contra-continuous function in topological spaces and established interesting results which related contracontinuity with compact spaces, S-closed spaces and strongly-S-closed spaces. The main purpose of this work is to introduce new variants of contracontinuous functions and characterize its by using Λ I -closed sets, in contrast with the variants of continuity studied by Sanabria et al [11]. Also, we study and investigate the preservation of several separation properties, connectedness and compactness, through direct and inverse images of such functions.…”

“…Suppose that (X, τ, I) is not a Λ I -connected space and each contra Λ I -continuous function f : (X, τ, I) → (Y, σ), where (Y, σ) is a T 0 -space, is a constant function. Since(X, τ, I) is not Λ Iconnected, by[11, Theorem 11], there exists a nonempty proper subset A of X which is both Λ Iopen and Λ I -closed. Let Y = {a, b}, σ = {Y, ∅, {a}, {b}} be a topology on Y and f : (X, τ, I) → (Y, σ, J) be a function such that f (A) = {a} and f (X − A) = {b}.…”