On Semi-Open Sets and Semi-Continuity

Abstract

In this paper, it is proved that if (X, T ) is a topological space, then the collection of all semi-open sets A in (X, T ) such that A ∩ B is semi-open for every semi-open set B in (X, T ) is a topology on X and that this topology is finer than the topology F (T ) constructed by S. Crossley in [1]. The π-relationship between these topologies is established. Characterizations of semi-continuous and irresolute maps are presented in terms of semi-limit and semi-closure.

On the position of abstract density topologies in the lattice of all topologies

“…As a nonempty semi open set, G has a nonempty interior in T 2 . The relation of similarity was investigated among others in [2] and [14] (under the name of π-relation). The relation of semi-correspondence introduced by N. Levine, was investigated also by S. G. Crossley and S. K. Hildebrand in [4] and T. R. Hamlett in [7].…”

“…He also defined the relation of semi-correspondence between topologies. Both definitions were used in the studies of semitopological properties (that means properties which are preserved under semi-homeomorphism, such as for instance separability, being T 2 , connectedness, see [4] for details), of semi-continuity ( [14]) and quasicontinuity ( [3]). A generalization of semi-open sets was discussed also by E. Ekici in [5].…”

On supra semi open sets and some applications on topological spaces

“…In 2008, Devi et al [7] studied supra α-open sets and α-continuous maps and in 2010, Sayed and Noiri [17] introduced and studied supra b-open sets and supra b-continuous maps. In 2012, Ramabhadrasarma and Srinivasakumar [15] constructed a new topology by utilizing a semi open sets notion and proved some characterizations of semi continuous maps. El-Shafei et al [8] presented and studied supra R-open sets and SR − T i -spaces (i = 1, 2).…”

Resolvability Properties of Similar Topologies

“…The notion of similarity between two topological spaces was introduced in [5], although the same relation was mentioned earlier in [16] as a π-relation. Among other results, [5] shows that every topology is similar to some abstract density topology.…”