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On some structures defined by means of neighbourhood assignments
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Type
Thesis
Author
Dewi Kartika Sari
Supervisor
Zhao, Dongsheng
Ho, Weng Kin
Abstract
In [72], Zhao introduced the concept of gauge compactness. In order to reveal deeper properties of gauge compactness, we propose a new cardinal function for an arbitrary topological space in Chapter 2; namely, the gauge compact index of a space. We are interested in spaces with finite gauge compact indices. We prove that a Hausdorff space with a finite gauge compact index is in fact a finite space. We obtain that if the gauge compact index of a T1 space X is 3 then X is hyperconnected. In this chapter, we also explore characterisations of M-uniformly continuous functions.
In Chapter 3, we introduce and investigate some classes of functions between two topological spaces by means of neighbourhood assignments. A function f : X → Y is called an AO-separated function if for any neighbourhood assignment ε on Y , there exists a neighbourhood assignment δ on X such that
(x1, x2) ∈ δ(x2) × δ(x1) ⇒ (f(x1), f(x2)) ∈ ε(f(x2)) × ε(f(x1)).
If X and Y are separable completely metrizable spaces, then an AO-separated function is exactly a Baire class one function. By considering various conditions on the codomain of a function, we show that there are many AO-separated functions that are Baire one functions when their codomains are metric spaces. This result gives a new characterisation for Baire class one functions between Polish spaces. In this part, we also introduce and study the notion of a near Baire one function.
In Chapter 4, we introduce and study some other topological properties by means of neighbourhood assignments. In this chapter, we introduce and investigate locally gauge compact spaces and we prove the following properties: (i) A space X is a quotient image of a locally gauge compact space if and only if for any A ⊆ X, the set A is open in X if and only if A∩K is open in K for every gauge compact K ⊆ X; and (ii) every Scott space of a poset is a quotient of a locally gauge compact space. In Section 3 of Chapter 4, we explore a covering properties using symmetric neighbourhood assignments, called s-compact. We prove that every compact space is s-compact, and every Hausdorff s-compact space is countably compact. Another concept that we introduce is the notion of completeness for a topological space.
One of methods to solve integral equations is using fixed point theorems. Since the Henstock- Kurzweil integral is more general than the Riemann integral and the Lebesgue integral, several mathematicians have been trying to solve integral equations using Henstock-Kurzweil integrals. However, the Denjoy space is not complete. Thus, it is natural to consider the problem: Can we define a complete topology?
Kurzweil said that the new convergence theorems for Henstock-Kurzweil integral involve only the primitives ([43]). In Chapter 5, following Kurzweil’s idea, we define some complete topologies on the space of primitives of Henstock-Kurzweil integrable functions instead of the Denjoy space. Furthermore, we introduce and explore a topology on the space L∞ and on the space BV of functions of bounded variation on [0, 1].
In Chapter 3, we introduce and investigate some classes of functions between two topological spaces by means of neighbourhood assignments. A function f : X → Y is called an AO-separated function if for any neighbourhood assignment ε on Y , there exists a neighbourhood assignment δ on X such that
(x1, x2) ∈ δ(x2) × δ(x1) ⇒ (f(x1), f(x2)) ∈ ε(f(x2)) × ε(f(x1)).
If X and Y are separable completely metrizable spaces, then an AO-separated function is exactly a Baire class one function. By considering various conditions on the codomain of a function, we show that there are many AO-separated functions that are Baire one functions when their codomains are metric spaces. This result gives a new characterisation for Baire class one functions between Polish spaces. In this part, we also introduce and study the notion of a near Baire one function.
In Chapter 4, we introduce and study some other topological properties by means of neighbourhood assignments. In this chapter, we introduce and investigate locally gauge compact spaces and we prove the following properties: (i) A space X is a quotient image of a locally gauge compact space if and only if for any A ⊆ X, the set A is open in X if and only if A∩K is open in K for every gauge compact K ⊆ X; and (ii) every Scott space of a poset is a quotient of a locally gauge compact space. In Section 3 of Chapter 4, we explore a covering properties using symmetric neighbourhood assignments, called s-compact. We prove that every compact space is s-compact, and every Hausdorff s-compact space is countably compact. Another concept that we introduce is the notion of completeness for a topological space.
One of methods to solve integral equations is using fixed point theorems. Since the Henstock- Kurzweil integral is more general than the Riemann integral and the Lebesgue integral, several mathematicians have been trying to solve integral equations using Henstock-Kurzweil integrals. However, the Denjoy space is not complete. Thus, it is natural to consider the problem: Can we define a complete topology?
Kurzweil said that the new convergence theorems for Henstock-Kurzweil integral involve only the primitives ([43]). In Chapter 5, following Kurzweil’s idea, we define some complete topologies on the space of primitives of Henstock-Kurzweil integrable functions instead of the Denjoy space. Furthermore, we introduce and explore a topology on the space L∞ and on the space BV of functions of bounded variation on [0, 1].
Date Issued
2019
Call Number
QA611.23 Dew