On α-continuity in topological spaces

Abstract

The aim of this paper is to give a systematic development of grill Ntopological spaces and discuss a few properties of local function. We build a topology for the corresponding grill by using the local function. Furthermore, we investigate the properties of weak forms of open sets in the grill N -topological spaces and discuss the relationships between them.2010 MSC: 54A05; 54A99; 54C10.

On faint continuity

“…The following fundamental relationship between classes of generalized open sets in a topological space was first proved in 1985 by Reilly and Vamanamurthy [16]. Nasef [11] has established by his set of Examples 4.1 to 4.5 and the diagram of his Remark 4.1 that faint α-continuity is distinct and independent from existing classes of functions.…”

On g-α-irresolute functions

“…Specifically, if τ X and τ Y are (ordinary) topologies on X and Y , then (τ X , τ Y )-continuity is classical topological continuity. Furthermore, (sτ X , τ Y )-continuity is the semi-continuity of [10], (pτ X , τ Y )-continuity is precontinuity in the sense of [11], (sτ X , sτ Y )-continuous functions are defined to be irresolute in [2], while (pτ X , pτ Y )-continuous functions are called preirresolute in [16]. So these are all examples of the use of the change of generalized topology technique to reduce a particular property of functions between ordinary topological spaces to generalized continuity.…”

Semi‐precontinuous functions and properties of generalized semi‐preclosed sets in topological spaces

“…Recall that a map f : X → Y is called M-preclosed [23] if the image of each preclosed set is a preclosed set.…”

“…In the literature the notions of semi [14] (resp., pre [17], α [18])-continuous mappings, semi [5] (resp., pre [17], α [18])-open mappings, semi [21] (resp., pre [12], α [18])-closed mappings, irresolute [10] (resp., preirresolute [23], α-irresolute [16])-mappings were introduced and studied using semi [14] (resp., pre [17], α [20])-open sets of and semi [6,9] (resp., pre [12], α [16,18])-closed subsets of X. In this paper, we introduce the concepts of semi-precontinuous mappings, semi-preopen mappings, semi-preclosed mappings, semi-preirresolute mappings, pre-semipreopen mappings, and pre-semi-preclosed mappings and study their characterizations in topological spaces using the semi-preopen sets and semi-preclosed sets due to D. Andrijevic [3] (note that β-open sets in [1] are the same as the semi-preopen sets of D. Andrijevic [3]).…”