Modification of generalized topologies via hereditary classes

Abstract

The purpose of the paper is to show that the construction leading from a topology and an ideal of sets to another topology remains valid, together with a lot of applications, if topology is replaced by generalized topology and ideal by hereditary class. IntroductionLet X be a set with power set exp X. Consider a topology τ on X and an ideal H on X, i.e. ∅ = H ⊂ exp X andIn the literature, there are a lot of papers based on a construction that, using τ and H, denes another topology τ * on X and discusses relations of τ and τ * (see e.g. [6]).

Characterizations of upper and lower \alpha(\mu_X,\mu_Y)-continuous multifunctions

“…Let B ⊆ P(X) satisfy ∅ ∈ B. Then all unions of some elements of B constitute a GT µ(B) and B is said to be a base for µ(B) [10]. Let µ be a GT on a set X = ∅.…”

A note on mathematical structures

“…A subfamily H ⊂ 2 X is called a hereditary class [4] if A ⊂ B, B ∈ H implies A ∈ H. This structure has been introduced byÁ. Császár in 2007 for the purpose of parallel study of ideal topological spaces [11,17].…”

\mu-\mu^{*} - Connectedness via Hereditary Classes

“…A generalized topology µ is called a quasi-topology [5] on X if U, V ∈ µ implies U ∩ V ∈ µ. A nonempty subset H of P (X) is called a hereditary class [6] …”

“…Csaszar [6] introduced the notion of generalized topological space with hereditary class. This is a generalization of an ideal topological space.…”