STUDY ON SOME TOPOLOGICAL CONCEPTS VIA IDEALS AND THEIR APPLICATIONS

Abstract

Throughout this thesis (X, τ), (Y, σ) and (Z, µ) (or simply X, Y and Z) are topological spaces with no separation axioms are assumed unless explicitly stated. For each A X, clτ(A) ( intτ(A)) denotes the closure (the interior) of A with respect to τ in order to avoid confusion when there exists more than one topology on X, otherwise the form cl(A) ( int(A)) will be used. Ƥ(A) the power set of A is the family of all subsets of A.

In 1933, Kuratowski [38] introduced the concept of an ideal on a nonempty set as follows: An ideal I on a nonempty set X is a nonempty collection of subsets of X which satisfies: (i) A∈ I and B A implies B∈ I, (hereditary), (ii) A∈ I and B∈ I implies A∪B∈ I, (finite additivity).

Also, several authors are interested in this line of study and therefore some sorts of ideals arise as one goes further in mathematics such as the ideal of finite subsets of X, the ideal of nowhere dense sets and the ideal of meager sets. Different types of operators in terms of ideals, compatibility property, compactness modulo an ideal, sets, functions and other concepts were introduced by many topologists.

The concept of the set operator ( )*: Ƥ(X) → Ƥ(X), called a local function, has been introduced by Vaidyanathaswamy in 1945 [68].

In 1967 [54] Newcomb defined a set operator Ѱ'(I, τ): τ → τ, on the other hand, in 1986, Natkaniec [53] defined another operator Ѱ(I,τ):Ƥ(X)→τ. It was shown in [23] that the operator Ѱ' is simply Ѱ restricted to τ.

In 1990, Hamlett, Rose and Jankovic' investigated many properties of the set operator ( )* in [24, 26, 31, 32, 61] and the operator Ѱ(I,τ) in [23]. Semadeni, in 1963 [64] and others have recognized a crucial property of ideals which was first proved for the ideal of meager sets by Banach in 1930 [13]. This property of ideals has been called "compatible" by Njastad in 1966 [56], Jankovic' and Hamlett in 1990 [26, 31], it was called super-compact by Vaidyanathaswamy in 1945 [68], "adherence" by Vaidyanathaswamy in 1960 [69] and strong Banach's localization property by Semadeni in 1963