Theta - ω - Mappings in Topological Spaces

Banach, Fréchet, Hilbert and Neumann Spaces

In 1943, Fomin [7] introduced the notion of θ-continuity. In 1966, the notions of θ-open subsets, θ-closed subsets and θ-closure were introduced by Veličko [18] for the purpose of studying the important class of H-closed spaces in terms of arbitrary filterbases. He also showed that the collection of θ-open sets in a topological space (X, τ) forms a topology on X denoted by τ θ (see also [12]). Dickman and Porter [4], [5], Joseph [11] continued the work of Veličko. Noiri and Jafari [15], Caldas et al. [1] and [2], Steiner [16] and Cao et al [3] have also obtained several new and interesting results related to these sets. In this paper, we will offer a finer topology on X than τ θ by utilizing the new notions of ω θ-open and ω θ-closed sets. We will also discuss some of the fundamental properties of such sets and some related maps.

2020

In this paper we present θ-semigeneralized pre-open maps, θ-semigeneralized pre-closed maps, pre-θ-semigeneralized pre-open maps, pre-θ-semigeneralized pre-closed maps by employing θ-sgp-closed sets. We also introduce higher separation axioms θsemigeneralized pre-regular spaces and θ-semigeneralized pre-normal spaces using θsgp-closed sets in topological spaces and we investigate some new characterizations of these mappings and higher separation axioms.

Russian Mathematical Surveys, 1980

Contemporary Mathematics

In this paper, new characterizations of pre-open (pre-closed, pre-continuous) maps will be obtained using the map η# induced via a map η between any two topological spaces. Further, these maps are characterized in terms of saturated sets under some sufficient conditions.

2017

In this paper we introduced RGW⍺LC-Continuous, RGW⍺LC*-Continuous, RGW⍺LC**-Continuous, sub-RGW⍺LC*-Continuous and RGW⍺LC-Irresolute Maps which are weaker than LC-Continuous and stronger than GβLC-Continuous and study some of their properties and their relationship with w-lc continuous, θ-lc-continuous, lδc-continuous and π-lc continuous etc.

Banach Journal of Mathematical Analysis, 2015

Let X and Y be topological spaces and f : X → Y be a continuous function. We are interested in finding points of Y at which f is open or closed. We will show that under certain conditions, the set of points of openness or closedness of f in Y , i. e. points of Y at which f is open (resp. closed) is a G δ subset of Y. We will extend some results of S. Levi, R. Engelking and I. A. Vaȋnšteȋn.

2004

In this paper we introduce the concept of (α, β, θ, ∂, I)-continuous mappings and prove that if α, β are operators on the topological space (X, τ) and θ, θ∗ , ∂ are operators on the topological space (Y, φ) and I a proper ideal on X, then a function f : X → Y is (α, β, θ ∧ θ∗, ∂, I)-continuous if and only if it is both (α, β, θ, ∂, I) -continuous and (α, β, θ∗, ∂, I)-continuous, generalizing a result of J. Tong. Additional results on (α, Int, θ, ∂, {∅})-continuous maps are given.

In this paper, we introduce and studies a new class of closed and open maps is called sαrw-closed and sαrw-open maps in topological space. Also some of their properties have been investigated. We also introduce sαrw*-closed and sαrw*-open maps in topological spaces and study of their properties.

atlas-conferences.com

The problem to characterize the continuity by images of sets was studied first in [V] by Velleman that have shown that the set C(R, R) of all real continuous functions on R can not be characterized by images of sets. ... [AP] FG Arenas, ML Puertas, Characterizing continuity in topological ...