Topological predomains and qcb spaces are not closed under sobrification (original) (raw)
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In Abstract Stone Duality the topology on a space X is treated, not as an infinitary lattice, but as an exponential space Σ X . This has an associated lambda calculus, in which monadicity of the self-adjunction Σ − Σ − makes all spaces sober and gives subspaces the subspace topology, and the Euclidean principle F σ ∧ σ = F ∧ σ makes Σ the classifier for open subspaces. Computably based locally compact locales provide the leading model for these axioms, although the methods are also applicable to CCD op (constructively completely distributive lattices).
Notre Dame Journal of Formal Logic, 1978
In this note we study Abraham Robinson's Q-topology and consider it as a means of constructing counter examples in topology. We shall be interested almost exclusively with separation and disconnectedness conditions in the Q-topology. For instance, we shall show that the Q-topology for a non-discrete completely regular space is a nondiscrete zero-dimensional space in every enlargement. The reader is assumed to know what is meant by an enlargement in the sense of Robinson, what is meant by an ultraproduct enlargement, and to be familiar with the rudiments of non-standard analysis. A good short introduction is sections 1-6 of [6]. We generalize the Q-topology somewhat by introducing the notion of a *topological space and the Q-topology for a *topological space. This will help in dealing with subspaces and will give a slightly simpler notation. Definition 1: A *topological space in a non-standard model *9W is a pair {X,%), where X is an internal set in *9W and$ ^P(X) is an internal family of sets closed under *finite intersections and internal unions, and which contains 0 and X. If (X,%) is a topological space in a model 3W, then *(X,$) = (*X,*$) is a (standard) *topological space in any enlargement *9W of 9W. If © is an infinite collection of topological spaces in a model 3W, then *© contains a non-standard *topological space for any enlargement *9W of 9W.
A Note on the Topologicity of Quantale-Valued Topological Spaces
Log. Methods Comput. Sci., 2017
For a quantale sfV{\sf{V}}sfV, the category sfV\sf VsfV-${\bf Top}$ of sfV{\sf{V}}sfV-valued topological spaces may be introduced as a full subcategory of those sfV{\sf{V}}sfV-valued closure spaces whose closure operation preserves finite joins. In generalization of Barr's characterization of topological spaces as the lax algebras of a lax extension of the ultrafilter monad from maps to relations of sets, for sfV{\sf{V}}sfV completely distributive, sfV{\sf{V}}sfV-topological spaces have recently been shown to be characterizable by a lax extension of the ultrafilter monad to sfV{\sf{V}}sfV-valued relations. As a consequence, sfV{\sf{V}}sfV-$\bf Top$ is seen to be a topological category over bfSet\bf SetbfSet, provided that sfV{\sf{V}}sfV is completely distributive. In this paper we give a choice-free proof that sfV{\sf{V}}sfV-$\bf Top$ is a topological category over bfSet\bf SetbfSet under the considerably milder provision that sfV{\sf{V}}sfV be a spatial coframe. When sfV{\sf{V}}sfV is a continuous lattice, that provision yields complete dis...
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Simple generators for the coreflective category of subsequential spaces, one of them countable, are constructed. EJery such must have subsequential order tiI. Subsequentialness is a local property and a countable property, both in a strong sense. A T2-subsequential space may be pseudocompact without being sequential, in contrast to T2-subsequential compact (countably compact, sequentially compact) spaces all being sequential. A compact subsequential space need not be sequential.
Hyperprojective Hierarchy of qcb0-Spaces
Lecture Notes in Computer Science, 2014
We extend the Luzin hierarchy of qcb 0 -spaces introduced in [ScS13] to all countable ordinals, obtaining in this way the hyperprojective hierarchy of qcb 0 -spaces. We generalize all main results of [ScS13] to this larger hierarchy. In particular, we extend the Kleene-Kreisel continuous functionals of finite types to the continuous functionals of countable types and relate them to the new hierarchy. We show that the category of hyperprojective qcb 0 -spaces has much better closure properties than the category of projective qcb 0 -space. As a result, there are natural examples of spaces that are hyperprojective but not projective.
Quotients of countably based spaces are not closed under sobrification
Mathematical Structures in Computer Science, 2006
In this note we show that quotients of countably based spaces (qcb spaces) and topological predomains as introduced by M. Schröder and A. Simpson are not closed under sobrification. As a consequence replete topological predomains need not be sober, i.e. in general repletion is not given by sobrification. Our counterexample also shows that a certain tentative "equalizer construction" of repletion fails for qcb spaces.
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Dado un ideal I y una topología τ sobre un conjunto X, existe una extensión natural τ I de τ asociada al ideal I que generaliza la extensión secuencial de τ . En este trabajo mostraremos algunas propiedades de τ I . Uno de los resultados principales dice que esta extensión preserva primero numerabilidad cuando el ideal I es un p-ideal analítico.
Some hierarchies of QCB 0-spaces
Mathematical Structures in Computer Science, 2014
We define and study hierarchies of topological spaces induced by the classical Borel and Luzin hierarchies of sets. Our hierarchies are divided into two classes: hierarchies of countably based spaces induced by their embeddings intoPω, and hierarchies of spaces (not necessarily countably based) induced by their admissible representations. We concentrate on the non-collapse property of the hierarchies and on the relationships between hierarchies in the two classes.