Continuity on Generalised Topological Spaces via Hereditary Classes

Abstract

A generalised topology is a collection of subsets of a given nonempty set containing the empty set and arbitrary unions of the elements in the collection. By using the concept of hereditary classes, a generalised topology can be extended to a new one, called a generalised topology via a hereditary class. We study continuity on generalised topological spaces via hereditary classes in various situations.

Theory of g-T_g-Maps

“…The concepts 1 of (T Ω , T Σ )-map π : T Ω → T Σ [1,2], g-(T Ω , T Σ )-map π g : T Ω → T Σ [9,18], (T g,Ω , T g,Σ )-map π : T g,Ω → T g,Σ [3], and g-(T g,Ω , T g,Σ )-map π g : T g,Ω → T g,Σ [21], called, respectively, ordinary and generalized maps (briefly, T-map and g-T-map, respectively) between T -spaces T Ω and T Σ , and ordinary and generalized maps (briefly, T g -map and g-T g -map, respectively) between T g -spaces T g,Ω and T g,Σ are all fundamental concepts that have been introduced and investigated by several mathematicians [12,16,17,20,21,23,29,31,33].…”

Generalized topologies induced by monotonic maps and hereditary classes

“…Following [5], we obtain some results of the continuity on generalized topological spaces via hereditary classes.…”

“…In 1990, Janković and Hamlett [4] defined a hereditary class H on a topological space X, which is the collection of subsets of X such that every subset of elements in H is in H. They also introduced the generalization of the closure in a topological space via the hereditary class H. In 2007, Császár [3] defined a new generalized topology which contains an old generalized topology via hereditary classes. In 2018, Montagantirud and Thaikua [5] introduced a notion of the continuity on generalized topological spaces via hereditary classes in various situations. They also proved that the continuity between two generalized topological spaces can be preserved on generalized topological spaces via hereditary classes.…”

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It is shown in [3] that every generalized topology can be extended to a new one by using hereditary classes, called a generalized topology via a hereditary class.Following [5], the continuity on generalized topological spaces via hereditary classes was studied. In this thesis, we introduce the notion of generalized topological spaces induced by monotonic maps and hereditary classes and provide some of their properties.

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“…Following [5], the continuity on generalized topological spaces via hereditary classes was studied. In this thesis, we introduce the notion of generalized topological spaces induced by monotonic maps and hereditary classes and provide some of their properties.…”