New Forms of μ-Compactness With Respect to Hereditary Classes

abstract: A hereditary class on a set X is a nonempty collection of subsets closed under heredity. The aim of this paper is to introduce and study strong forms of μ-compactness in generalized topological spaces with respect to a hereditary class, called SμH-compactness and S− SμH-compactness. Also several of their properties are presented. Finally some effects of various kinds of functions on them are studied.


Introduction
This work is developed around the concept of µ-compactness with respect a hereditary class which was introduced by Carpintero, Rosas, Salas-Brown and Sanabria in [4].In this research, we use the notions of generalized topology and hereditary class introduced by Császár in [1] and [2], respectively, in order to define and characterize the SµH-compactness and S−SµH-compactness spaces.Also some properties of these spaces are obtained and the behavior of these spaces under certain kinds of functions also is investigated.The strategy of using generalized topologies and hereditary classes to extend classical topological concepts have been used by many authors such as [2], [6], [9], [14], among others..

Preliminaries
Let X be a non-empty set and 2 X denote the power set of X.We call a class µ ⊆ 2 X a generalized topology [1] (briefly, GT) if ∅ ∈ µ and arbitrary union of elements of µ belongs to µ.A set X with a GT is called a generalized topological space (briefly, GTS) and is denoted by (X, µ).For a GTS (X, µ), the elements of µ are called µ-open sets and the complement of µ-open sets are called µ-closed sets.For A ⊆ X, we denote by c µ (A) the intersection of all µ-closed sets containing A,

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A. Qahis i.e., the smallest µ-closed set containing A and by i µ (A) the union of all µ-open sets contained in A, i.e., the largest µ-open set contained in A (see [1], [3] [5].A subset A of X is said to be µ-compact if for every µ-covering {V α : α ∈ Λ} of A there exists a finite subfamily {V α : α ∈ Λ 0 } that also covers A. X is said to be µ-compact if X is µ-compact as a subset [5]. A nonempty family H of subsets of X is called a hereditary class [2] if A ∈ H and B ⊂ A imply that B ∈ H.Given a generalized topological space (X, µ) with a hereditary class H, for a subset A of X, the generalized local function of A with respect to H and µ [2] is defined as follows: A * = {x ∈ X : U ∩ A / ∈ H for all U ∈ µ x }, where µ x = {U : x ∈ U and U ∈ µ}.And for A a subset of X, is defined: then H is called an ideal on X [7].We call (X, µ, H) a hereditary generalized topological space and briefly we denote it by HGTS.If (X, µ, H) is a HGTS, the set B = {V \ H : V ∈ µ and H ∈ H} is a base for a GT µ * , finer than µ [2].If there is no confusion, we simply write Theorem 2.4.[2] Let (X, µ) be a GTS and H be a hereditary class on X and A a subset of X, then

SµH-Compact Spaces
We recall that a subset A of a HGTS (X, µ, H) is said to be µH-compact [4], if for every µ-open cover {V α : α ∈ Λ} of A by elements of µ, there exists a finite subset Λ 0 of Λ such that A\ α∈Λ0 V α ∈ H.The HGTS (X, µ, H) is said to be µH-compact if X is µH-compact as a subset.Definition 3.1.Let (X, µ) be a GT S and H be a hereditary class on X.A subset A of X is said to be strong µH-compact (briefly SµH-compact) if for every family New Forms of µ-Compactness With Respect to Hereditary Classes 23 The converse is not true as shown by the following example.
We note that if A is µH g -closed then A is µg-closed.The converse is not true as shown by the following examples.Proposition 3.6.Let (X, µ, H) be a HGTS and B be a base for µ.Then the following are equivalent: 1. (X, µ, H) is SµH-compact;
(2) ⇒ (1): Let {V α : α ∈ Λ} be a family of non-empty µ-open subsets of X such that X \ α∈Λ V α ∈ H.For each α ∈ Λ there exists a family {B αβ : and by ( 2) there exist H) is a HGTS then the following are equivalent: 1. (X, µ, H) is SµH-compact; H) is a HGTS and H is an ideal, then the following are equivalent: For some x ∈ X, there exists α x ∈ Λ such that x ∈ V α x .Then there exist U α x ∈ µ x and (2) ⇒ (1): It is obvious.✷

For any family {F
Next we study the behavior of some types of subsets of a SµH-compact set of X.
So A ∪ B is SµH-compact.✷ The following example shows that the previous theorem does not hold when H is just a hereditary class, not an ideal.