Base-Complexity Classifications of QCB0-Spaces

We define and study new classifications of qcb\(_0\)-spaces based on the idea to measure the complexity of their bases. The new classifications complement those given by the hierarchies of qcb\(_0\)-spaces introduced in [7, 8] and provide new tools to investigate non-countably based qcb\(_0\)-spaces. As a by-product, we show that there is no universal qcb\(_0\)-space and establish several apparently new properties of the Kleene-Kreisel continuous functionals of countable types.

[1]  K. Hofmann,et al.  A Compendium of Continuous Lattices , 1980 .

[2]  Victor L. Selivanov,et al.  Some hierarchies of QCB 0-spaces , 2013, Mathematical Structures in Computer Science.

[3]  Camillo Costantini,et al.  On the dissonance of some metrizable spaces , 1998 .

[4]  Matthias Schröder,et al.  Extended admissibility , 2002, Theor. Comput. Sci..

[5]  D. Normann.  The countable functionals , 1980 .

[6]  Matthias Schröder,et al.  Spaces allowing Type‐2 Complexity Theory revisited , 2004, Math. Log. Q..

[7]  Thomas Streicher,et al.  Quotients of countably based spaces are not closed under sobrification , 2006, Math. Struct. Comput. Sci..

[8]  Victor L. Selivanov,et al.  Towards a descriptive set theory for domain-like structures , 2006, Theor. Comput. Sci..

[9]  D. Dalen.  Review: Georg Kreisel, Godel's Intepretation of Heyting's Arithmetic; G. Kreisel, Relations Between Classes of Constructive Functionals; Georg Kreisel, A. Heyting, Interpretation of Analysis by Means of Constructive Functionals of Finite Types , 1971 .

[10]  Szymon Dolecki,et al.  When do the upper Kuratowski topology (homeomorphically, Scott topology) and the co-compact topology coincide? , 1995 .

[11]  Victor L. Selivanov,et al.  Some Hierarchies of QCB0-Spaces , 2013, ArXiv.

[12]  Nitakshi Goyal,et al.  General Topology-I , 2017 .

[13]  V. L. Selivanov,et al.  Difference Hierarchy in ϕ-Spaces , 2004 .

[14]  Ahmed Bouziad,et al.  Consonance and topological completeness in analytic spaces , 1999 .

[15]  Christoph Kreitz,et al.  Theory of Representations , 1985, Theor. Comput. Sci..

[16]  Victor L. Selivanov,et al.  Total Representations , 2013, Log. Methods Comput. Sci..

[17]  Klaus Weihrauch,et al.  Computable Analysis: An Introduction , 2014, Texts in Theoretical Computer Science. An EATCS Series.

[18]  Matthias Schröder,et al.  Two preservation results for countable products of sequential spaces , 2007, Mathematical Structures in Computer Science.

[19]  Benjamin Naumann,et al.  Classical Descriptive Set Theory , 2016 .

[20]  Dmitri Shakhmatov,et al.  When does the Fell topology on a hyperspace of closed sets coincide with the meet of the upper Kuratowski and the lower Vietoris topologies , 1996 .

[21]  Douglas S. Bridges,et al.  Constructivity in Mathematics , 2004 .

[22]  Victor L. Selivanov,et al.  Hyperprojective hierarchy of qcb0-spaces , 2014, De Computis.

[23]  A. H. Stone,et al.  Quotients of the space of irrationals , 1969 .

[24]  Andrea Schalk,et al.  Algebras for generalized power constructions , 1993 .

[25]  Peter Hertling,et al.  Topological properties of real number representations , 2002, Theor. Comput. Sci..

[26]  Matthew de Brecht.  Quasi-Polish spaces , 2011, Ann. Pure Appl. Log..

[27]  S. Shelah,et al.  Annals of Pure and Applied Logic , 1991 .

[28]  Matthias Schröder,et al.  Admissible representations for continuous computations , 2003 .

[29]  Matthias Schröder.  A Hofmann-Mislove Theorem for Scott open sets , 2015 .

[30]  M. Smyth.  Power Domains and Predicate Transformers: A Topological View , 1983, ICALP.

[31]  J. Lawson,et al.  Comparing Cartesian closed categories of (core) compactly generated spaces , 2004 .