Boolean Images Of Connected Spaces

Al Safe, Arwa (2022) Boolean Images Of Connected Spaces. Doctoral thesis, University of East Anglia.

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This thesis investigates an interesting generalisation of the concept of continuous functions, namely the notion of Boolean image. This type of image does not preserve connectedness, but otherwise has many of the properties of continuous images. We analyse this notion on many different kinds of topological spaces, deepening our understanding of combinatorial methods from set theory. The main contributions of this thesis can be summarized as follows. In chapter 3, we prove that every compact subspace of 2k with a finite support is a Boolean image of a connected space. This main result is followed by some applications in chapters 4 and 5 to different spaces, such as Eberlein compact spaces and Radon-Nikodým spaces, respectively. Chapter 6 centres around the connection between Banach spaces of continuous functions C(K) and C(L), in the case that the spaces L and K are both compact and zero-dimensional. In particular, we prove that if L is a a bijective Boolean image of a compact zero-dimensional space K, then C(L) is isometric to C(K). Moreover, we prove that if L is a Boolean image of K, then C(L) is isometric to a subspace of C(K). On the other hand, we prove that if the Banach spaces C(K) and C(L) are isomorphic, where the spaces K and L are zero-dimensional, then there is a subspace K′ in K and a subspace L′ in L such that K′ is a Boolean image of L′. In chapter 7, we examine some cardinal functions in terms of the possibility of being transferred via a Boolean image. In this respect, we show that weight and countable density are preserved by Boolean images.

Item Type: Thesis (Doctoral)
Faculty \ School: Faculty of Science > School of Mathematics
Depositing User: Nicola Veasy
Date Deposited: 29 Nov 2022 12:17
Last Modified: 29 Nov 2022 12:17

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