Petrakis, Iosif (2015): Constructive topology of bishop spaces. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics 

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Abstract
The theory of Bishop spaces (TBS) is so far the least developed approach to constructive topology with points. Bishop introduced function spaces, here called Bishop spaces, in 1967, without really exploring them, and in 2012 Bridges revived the subject. In this Thesis we develop TBS. Instead of having a common spacestructure on a set X and R, where R denotes the set of constructive reals, that determines a posteriori which functions of type X > R are continuous with respect to it, within TBS we start from a given class of "continuous" functions of type X > R that determines a posteriori a spacestructure on X. A Bishop space is a pair (X, F), where X is an inhabited set and F, a Bishop topology, or simply a topology, is a subset of all functions of type X > R that includes the constant maps and it is closed under addition, uniform limits and composition with the Bishop continuous functions of type R > R. The main motivation behind the introduction of Bishop spaces is that functionbased concepts are more suitable to constructive study than setbased ones. Although a Bishop topology of functions F on X is a set of functions, the settheoretic character of TBS is not that central as it seems. The reason for this is Bishop's inductive concept of the least topology generated by a given subbase. The definitional clauses of a Bishop space, seen as inductive rules, induce the corresponding induction principle. Hence, starting with a constructively acceptable subbase the generated topology is a constructively graspable set of functions exactly because of the corresponding principle. The functiontheoretic character of TBS is also evident in the characterization of morphisms between Bishop spaces. The development of constructive pointfunction topology in this Thesis takes two directions. The first is a purely topological one. We introduce and study, among other notions, the quotient, the pointwise exponential, the dual, the Hausdorff, the completely regular, the 2compact, the paircompact and the 2connected Bishop spaces. We prove, among other results, a StoneCech theorem, the Embedding lemma, a generalized version of the Tychonoff embedding theorem for completely regular Bishop spaces, the GelfandKolmogoroff theorem for fixed and completely regular Bishop spaces, a StoneWeierstrass theorem for pseudocompact Bishop spaces and a StoneWeierstrass theorem for paircompact Bishop spaces. Of special importance is the notion of 2compactness, a constructive functiontheoretic notion of compactness for which we show that it generalizes the notion of a compact metric space. In the last chapter we initiate the basic homotopy theory of Bishop spaces. The other direction in the development of TBS is related to the analogy between a Bishop topology F, which is a ring and a lattice, and the ring of realvalued continuous functions C(X) on a topological space X. This analogy permits a direct "communication" between TBS and the theory of rings of continuous functions, although due to the classical settheoretic character of C(X) this does not mean a direct translation of the latter to the former. We study the zero sets of a Bishop space and we prove the Urysohn lemma for them. We also develop the basic theory of embeddings of Bishop spaces in parallel to the basic classical theory of embeddings of rings of continuous functions and we show constructively the Urysohn extension theorem for Bishop spaces. The constructive development of topology in this Thesis is within Bishop's informal system of constructive mathematics BISH, inductive definitions with rules of countably many premises included.
Item Type:  Theses (Dissertation, LMU Munich) 

Keywords:  constructive mathematics, constructive topology, Bishop spaces 
Subjects:  500 Natural sciences and mathematics 500 Natural sciences and mathematics > 510 Mathematics 
Faculties:  Faculty of Mathematics, Computer Science and Statistics 
Language:  English 
Date of oral examination:  13. July 2015 
1. Referee:  Schwichtenberg, Helmut 
MD5 Checksum of the PDFfile:  576dfd8c115fba678c0f7c8665dfa714 
Signature of the printed copy:  0001/UMC 23326 
ID Code:  18822 
Deposited On:  03. Nov 2015 12:22 
Last Modified:  23. Oct 2020 21:29 