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Metric Spaces in Synthetic Topology
, 2010
"... We investigate the relationship between the synthetic approach to topology, in which every set is equipped with an intrinsic topology, and constructive theory of metric spaces. We relate the synthetic notion of compactness of Cantor space to Brouwer’s Fan Principle. We show that the intrinsic and me ..."
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We investigate the relationship between the synthetic approach to topology, in which every set is equipped with an intrinsic topology, and constructive theory of metric spaces. We relate the synthetic notion of compactness of Cantor space to Brouwer’s Fan Principle. We show that the intrinsic and metric topologies of complete separable metric spaces coincide if they do so for Baire space. In Russian Constructivism the match between synthetic and metric topology breaks down, as even a very simple complete totally bounded space fails to be compact, and its topology is strictly finer than the metric topology. In contrast, in Brouwer’s intuitionism synthetic and metric notions of topology and compactness agree. 1
The intrinsic topology of a MartinLöf universe
, 2012
"... Assuming the propositional axiom of extensionality, we show that a MartinLöf universe à la Russell is indiscrete in its intrinsic topology. This doesn’t invoke Brouwerian continuity principles. As a corollary, we derive Rice’s Theorem for the universe: the existence of a nontrivial, decidable, ext ..."
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Assuming the propositional axiom of extensionality, we show that a MartinLöf universe à la Russell is indiscrete in its intrinsic topology. This doesn’t invoke Brouwerian continuity principles. As a corollary, we derive Rice’s Theorem for the universe: the existence of a nontrivial, decidable, extensional property of the universe implies the weak limited principle of omniscience. This is a theorem in type theory. Without assuming extensionality, we deduce the following metatheorem: in intensional MartinLöf type theory with a universe, there is no closed term defining a nontrivial, decidable, extensional property of the universe. 1
A HofmannMislove theorem for bitopological spaces
, 2008
"... We present a Stone duality for bitopological spaces in analogy to the duality between topological spaces and frames, and discuss the resulting notions of sobriety and spatiality. Under the additional assumption of regularity, we prove a characterisation theorem for subsets of a bisober space that ar ..."
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We present a Stone duality for bitopological spaces in analogy to the duality between topological spaces and frames, and discuss the resulting notions of sobriety and spatiality. Under the additional assumption of regularity, we prove a characterisation theorem for subsets of a bisober space that are compact in one and closed in the other topology. This is in analogy to the celebrated HofmannMislove theorem for sober spaces. We link the characterisation to Taylor’s and Escardó’s reading of the HofmannMislove theorem as continuous quantification over a subspace.
cfl2004 Published by Elsevier
"... 2 Abstract. Synthetic topology as conceived in this monograph has three fundamental aspects: 1. to explain what has been done in classical topology in conceptual terms, 2. to provide oneline, enlightening proofs of the theorems that constitute the core of the theory, and 3. to smoothly export topol ..."
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2 Abstract. Synthetic topology as conceived in this monograph has three fundamental aspects: 1. to explain what has been done in classical topology in conceptual terms, 2. to provide oneline, enlightening proofs of the theorems that constitute the core of the theory, and 3. to smoothly export topological concepts and theorems to unintended situations, keeping the synthetic proofs unmodified.
Locality, Weak or Strong Anticipation and Quantum Computing. I. Nonlocality in Quantum Theory
"... Abstract The universal Turing machine is an anticipatory theory of computability by any digital or quantum machine. However the ChurchTuring hypothesis only gives weak anticipation. The construction of the quantum computer (unlike classical computing) requires theory with strong anticipation. Categ ..."
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Abstract The universal Turing machine is an anticipatory theory of computability by any digital or quantum machine. However the ChurchTuring hypothesis only gives weak anticipation. The construction of the quantum computer (unlike classical computing) requires theory with strong anticipation. Category theory provides the necessary coordinatefree mathematical language which is both constructive and nonlocal to subsume the various interpretations of quantum theory in one pullback/pushout Dolittle diagram. This diagram can be used to test and classify physical devices and proposed algorithms for weak or strong anticipation. Quantum Information Science is more than a merger of ChurchTuring and quantum theories. It has constructively to bridge the nonlocal chasm between the weak anticipation of mathematics and the strong anticipation of physics.
Aspects of slice stability in Locale Theory
, 2008
"... It is shown that a particular categorical axiomatisation of the category of locales is slice stable. This localic slice stability can be used to recover the fundamental theorem of topos theory. A categorical account of the ideal completion of a preorder is developed and is used to give a new proof o ..."
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It is shown that a particular categorical axiomatisation of the category of locales is slice stable. This localic slice stability can be used to recover the fundamental theorem of topos theory. A categorical account of the ideal completion of a preorder is developed and is used to give a new proof of Joyal and Tierney’s result on the slice stability of locales.