Studies on topological ordered spaces
Studies on topological ordered spaces
The study of topological ordered spaces was initiated by Nachbin [15]. In this thesis we are interested in extending Nachbin's work; our main concern is with the problem of representing ordered spaces, which have such properties as local compactness, or regularity and other suitable conditions, into countable products Iω of copies of the unit interval I with the vector order. We also show how to embed a locally compact ordered space in a contractible and locally contractible topological sublattice of Iω. By contrast with the situation in the topological category T we give possibly surprising examples in the category of topological ordered spaces TO. The first is an ordering of I which cannot be order-embedding in In for any finite n. The second example shows a second countable metric ordered space that is not order-embeddable in Iω. All of these results appear in Chapter 4, and they depend on material introduced in the earlier chapters. Thus Chapter 1 and 2 contain basic notation, abstract theorems on closure operators on lattices, and material on the extension of morphisms. Modified order-separation axioms are leading to questions of normal-order and para-compact-order investigated in Chapter 3. Finally in Chapter 5 we introduce semi-metrisable ordered spaces and discuss order completions; our approach is quite natural via the semi-uniform structure rather than by Redfields' approach using nets [18]; we conclude with a detailed study of order completion in Q, the rational number with the Euclidean topology and the vector order. (D75035/87)
University of Southampton
Nada, Shoukry Ibrahim Mohmed
1986
Nada, Shoukry Ibrahim Mohmed
Nada, Shoukry Ibrahim Mohmed
(1986)
Studies on topological ordered spaces.
University of Southampton, Doctoral Thesis.
Record type:
Thesis
(Doctoral)
Abstract
The study of topological ordered spaces was initiated by Nachbin [15]. In this thesis we are interested in extending Nachbin's work; our main concern is with the problem of representing ordered spaces, which have such properties as local compactness, or regularity and other suitable conditions, into countable products Iω of copies of the unit interval I with the vector order. We also show how to embed a locally compact ordered space in a contractible and locally contractible topological sublattice of Iω. By contrast with the situation in the topological category T we give possibly surprising examples in the category of topological ordered spaces TO. The first is an ordering of I which cannot be order-embedding in In for any finite n. The second example shows a second countable metric ordered space that is not order-embeddable in Iω. All of these results appear in Chapter 4, and they depend on material introduced in the earlier chapters. Thus Chapter 1 and 2 contain basic notation, abstract theorems on closure operators on lattices, and material on the extension of morphisms. Modified order-separation axioms are leading to questions of normal-order and para-compact-order investigated in Chapter 3. Finally in Chapter 5 we introduce semi-metrisable ordered spaces and discuss order completions; our approach is quite natural via the semi-uniform structure rather than by Redfields' approach using nets [18]; we conclude with a detailed study of order completion in Q, the rational number with the Euclidean topology and the vector order. (D75035/87)
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Published date: 1986
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Local EPrints ID: 461210
URI: http://eprints.soton.ac.uk/id/eprint/461210
PURE UUID: 6147b376-6887-4f4d-84eb-8ea31aa33656
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Date deposited: 04 Jul 2022 18:38
Last modified: 04 Jul 2022 18:38
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Author:
Shoukry Ibrahim Mohmed Nada
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