Maximilian Ganster

Bio: Maximilian Ganster is an academic researcher from Graz University of Technology. The author has contributed to research in topics: Closed set & Topological space. The author has an hindex of 10, co-authored 34 publications receiving 469 citations. Previous affiliations of Maximilian Ganster include University of Graz.

Locally closed sets and LC-continuous functions

Abstract: In this paper we introduce and study three different notions of generalized continuity, namely LC-irresoluteness, LC-continuity and sub-LC-continuity. All three notions are defined by using the concept of a locally closed set. A subset S of a topological space X is locally closed if it is the intersection of an open and a closed set. We discuss some properties of these functions and show that a function between topological spaces is continuous if and only if it is sub-LC-continuous and nearly continuous in the sense of Ptak. Several examples are provided to illustrate the behavior of these new classes of functions.

A decomposition of continuity

Abstract: In 1922 Blumberg[1] introduced the notion of a real valued function on Euclidean space being densely approached at a point in its domain. Continuous functions satisfy this condition at each point of their domains. This concept was generalized by Ptak[7] in 1958 who used the term ’nearly continuous’, and by Husain[3] in 1966 under the name of ’almost continuity’. More recently, Mashhour et al. [5] have called this property of functions between arbitrary topological spaces ’precontinuity’. In this paper we define a new property of functions between topological spaces which is the dual of Blumberg’s original notion, in the sense that together they are equivalent to continuity. Thus we provide a new decomposition of continuity in Theorem 4 (iv) which is of some historical interest. In a recent paper [10] , Tong introduced the notion of an A–set in a topological space and the concept of A–continuity of functions between topological spaces. This enabled him to produce a new decomposition of continuity. In this paper we improve Tong’s decomposition result and provide a decomposition of A–continuity.

Preopen sets and resolvable spaces

Abstract: K1: Find necessary and sufficient conditions under which every preopen set is open. K2: Find conditions under which every dense-in-itself set is preopen. K3: Find conditions under which the intersection of any two preopen sets preopen. K4: If (X, τ) is a topological space, let τ ∗ denote the topology on X obtained by taking the collection of all preopen sets of (X, τ) as a subbase. Find conditions under which τ ∗ is discrete. ∗AMS Subject Classification : 54 A 05, 54 A 10;

in Topological Spaces

Abstract: 1. Definition of a topological space Definition 1.1. A topology T on a set X is a collection of subsets of X subject to the following three rules, called the axioms of a topology: 1. The empty set ∅ and all of X belong to T. 2. If U i , i ∈ I is a collection of elements from T (indexed by the set I which may well be infinite), then ∪ i∈I U i must belong to T as well. Said differently, T is closed under taking arbitrary unions. must also belong to T. Thus we demand that T be closed under taking finite intersections. A topological space is a pair (X, T) consisting of a set X and a topology T on X. A subset U of X is called an open set if U ∈ T and a subset V ⊂ X is called closed if X − V is open. A neighborhood of a point x ∈ X is any open set U ⊂ X that contains x. This definition is central to the remainder of the book and so, before moving on to consider examples, we first pause to elucidate its various aspects. The choice of the three axioms of a topology should not be too surprising given the results from chapter ??. Specifically, they are modeled on the three properties of open subsets of Euclidean space proved in theorem ??. Keeping in mind that open subsets of R n were used to recast the definition of continuity (see theorem ??), the attentive reader will have little difficulty in guessing what the definition of a continuous function between two topological spaces should be (for an answer, see definition ??). Given a topological space (X, T) we shall often simply write X when the topology T is understood from context, and refer to X as a topological space. On the other hand, when several topological spaces are involved in a discussion, we may label the topology T by T X to indicate that it belongs with X. For example, we shall write (X, T X) or (Y, T Y) to label topological spaces. The notion of open and closed subset of a topological space X shall be crucial to all subsequent chapters. Whether or not a given subset A ⊂ X is open or not, depends on the choice of a topology T on X. …

Contra-continuous functionsand strongly s-closed spaces

Abstract: In 1989 Ganster and Reilly [6] introduced and studied the notion of LC-continuous functions via the concept of locally closed sets. In this paper we consider a stronger form of LC-continuity called contra-continuity. We call a function f:(X,τ)→(Y,σ) contra-continuous if the preimage of every open set is closed. A space (X,τ) is called strongly S-closed if it has a finite dense subset or equivalently if every cover of (X,τ) by closed sets has a finite subcover. We prove that contra-continuous images of strongly S-closed spaces are compact as well as that contra-continuous, β-continuous images of S-closed spaces are also compact. We show that every strongly S-closed space satisfies FCC and hence is nearly compact.

Nonsymmetric Distances and Their Associated Topologies: About the Origins of Basic Ideas in the Area of Asymmetric Topology

Abstract: We begin with some remarks explaining the structure of this article. After some introductory statements in the following paragraphs, we summarize the historic development of what is now often called “Nonsymmetric or Asymmetric Topology” in Section 2. In the following, more specific sections we discuss the historic development of some of the main ideas of the area in greater detail. The list of sections and keywords given above should help the specialist to find his way through the various sections.

An Extension of Neutrosophic AHP–SWOT Analysis for Strategic Planning and Decision-Making

Abstract: Every organization seeks to set strategies for its development and growth and to do this, it must take into account the factors that affect its success or failure. The most widely used technique in strategic planning is SWOT analysis. SWOT examines strengths (S), weaknesses (W),opportunities (O) and threats (T), to select and implement the best strategy to achieve organizational goals.