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COMPUTABLY COMPACT METRIC SPACES

Published online by Cambridge University Press:  11 May 2023

RODNEY G. DOWNEY
Affiliation:
SCHOOL OF MATHEMATICS AND STATISTICS VICTORIA UNIVERSITY OF WELLINGTON WELLINGTON, NEW ZEALAND E-mail: Rod.Downey@msor.vuw.ac.nz E-mail: alexander.g.melnikov@gmail.com E-mail: sasha.melnikov@vuw.ac.nz
ALEXANDER G. MELNIKOV
Affiliation:
SCHOOL OF MATHEMATICS AND STATISTICS VICTORIA UNIVERSITY OF WELLINGTON WELLINGTON, NEW ZEALAND E-mail: Rod.Downey@msor.vuw.ac.nz E-mail: alexander.g.melnikov@gmail.com E-mail: sasha.melnikov@vuw.ac.nz

Abstract

We give a systematic technical exposition of the foundations of the theory of computably compact metric spaces. We discover several new characterizations of computable compactness and apply these characterizations to prove new results in computable analysis and effective topology. We also apply the technique of computable compactness to give new and less combinatorially involved proofs of known results from the literature. Some of these results do not have computable compactness or compact spaces in their statements, and thus these applications are not necessarily direct or expected.

Type
Articles
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

Aberth, O., Computable Analysis , McGraw-Hill, New York, 1980.Google Scholar
Alexandroff, P., Untersuchungen über gestalt und Lage Abgeschlossener Mengen Beliebiger dimension . Annals of Mathematics. Second Series , vol. 30 (1928/29), nos. 1–4, pp. 101187.10.2307/1968272CrossRefGoogle Scholar
Ash, C. and Knight, J., Computable Structures and the Hyperarithmetical Hierarchy , Studies in Logic and the Foundations of Mathematics, vol. 144, North-Holland, Amsterdam, 2000.Google Scholar
Avigad, J. and Brattka, V., Computability and analysis: The legacy of Alan Turing , Turing’s Legacy: Developments from Turing’s Ideas in Logic (R. Downey, editor), Lecture Notes in Logic, vol. 42, Association for Symbolic Logic, La Jolla, 2014, pp. 147.Google Scholar
Bagaviev, R., Batyrshin, I., Bazhenov, N., Bushtets, D., Dorzhieva, M., Kornev, R., Melnikov, A., Ng, K. M., and Koh, H. T., Computably and punctually universal spaces, preprint, 2021.Google Scholar
Barrett, J., Downey, R., and Greenberg, N., Cousin’s lemma in second-order arithmetic. Proceedings of the American Mathematical Society Series B , vol. 9 (2022), pp. 111124.Google Scholar
Bazhenov, N., Downey, R., Kalimullin, I., and Melnikov, A., Foundations of online structure theory, this Journal, vol. 25 (2019), no. 2, pp. 141181.Google Scholar
Bazhenov, N., Harrison-Trainor, M., and Melnikov, A., Computable Stone spaces, preprint, 2021, arXiv:2107.01536.Google Scholar
Ben Yaacov, I., Berenstein, A., Henson, C. W., and Usvyatsov, A., Model theory for metric structures , Model Theory with Applications to Algebra and Analysis , vol. 2 (Z. Chatzidakis, D. Macpherson, A. Pillay, and A. Wilkie, editors), London Mathematical Society Lecture Note series, vol. 350, Cambridge University Press, Cambridge, 2008, pp. 315427.10.1017/CBO9780511735219.011CrossRefGoogle Scholar
Ben Yaacov, I. and Pedersen, A. P., A proof of completeness for continuous first-order logic . Journal of Symbolic Logic , vol. 75 (2010), no. 1, pp. 168190.10.2178/jsl/1264433914CrossRefGoogle Scholar
Bishop, E., Foundations of Constructive Analysis , McGraw-Hill, New York, 1967.Google Scholar
Borel, E., Le calcul des intégrales définies . Journal de Mathématiques Pures et Appliquées , vol. 8 (1912), pp. 159210.Google Scholar
Bosserhoff, V., On the effective existence of Schauder bases . Journal of Universal Computer Science , vol. 15 (2009), no. 6, pp. 11451161.Google Scholar
Bosserhoff, V. and Hertling, P., Effective subsets under homeomorphisms of ${\mathbb{R}}^n$ . Information and Computation , vol. 245 (2015), pp. 197212.CrossRefGoogle Scholar
Brattka, V., Computability of Banach Space Principles , FernUniversität Hagen, Fachbereich Informatik, Hagen, 2001.Google Scholar
Brattka, V., Borel complexity and computability of the Hahn–Banach theorem . Archive for Mathematical Logic , vol. 46 (2008), nos. 7–8, pp. 547564.CrossRefGoogle Scholar
Brattka, V., Plottable real number functions and the computable graph theorem . SIAM Journal on Computing , vol. 38 (2008), no. 1, pp. 303328.10.1137/060658023CrossRefGoogle Scholar
Brattka, V. and Dillhage, R.. Computability of compact operators on computable Banach spaces with bases . Mathematical Logic Quarterly , vol. 53 (2007), nos. 4–5, pp. 345364.CrossRefGoogle Scholar
Brattka, V. and Presser, G., Computability on subsets of metric spaces . Theoretical Computer Science , vol. 305 (2003), nos. 1–3, pp. 4376.CrossRefGoogle Scholar
Brattka, V., de Brecht, M., and Pauly, A., Closed choice and a uniform low basis theorem . Annals of Pure and Applied Logic , vol. 163 (2012), no. 8, pp. 9861008.10.1016/j.apal.2011.12.020CrossRefGoogle Scholar
Brattka, V., Le Roux, S., Miller, J. S., and Pauly, A., Connected choice and the Brouwer fixed point theorem . Journal of Mathematical Logic , vol. 19 (2019), no. 1, 1950004, 46 pp.CrossRefGoogle Scholar
Braverman, M. and Yampolsky, M., Computability of Julia Sets , Algorithms and Computation in Mathematics, vol. 23, Springer, Berlin, 2009.Google Scholar
de Brecht, M., Pauly, A., and Schröder, M., Overt choice . Computability , vol. 9 (2020), nos. 3–4, pp. 169191.CrossRefGoogle Scholar
Burnik, K. and Iljazović, Z., Computability of 1-manifolds . Logical Methods in Computer Science , vol. 10 (2014), no. 2:8, pp. 128.Google Scholar
Camrud, C., Goldbring, I., and McNicholl, T. H., On the complexity of the theory of a computably presented metric structure. Archive for Mathematical Logic , to appear, preprint, 2021, arXiv:2106.05372.Google Scholar
Carothers, N. L., A Short Course on Banach Space Theory , London Mathematical Society Student Texts, vol. 64, Cambridge University Press, Cambridge, 2005.Google Scholar
Ceitin, G. S., Algorithmic operators in constructive complete separable metric spaces . Doklady Akademii Nauk SSSR , vol. 128 (1959), pp. 4952.Google Scholar
Čelar, M. and Iljazović, Z., Computability of products of chainable continua . Theory of Computing Systems , vol. 65 (2021), no. 2, pp. 410427.CrossRefGoogle Scholar
Cenzer, D., ${\varPi}_1^0$ classes in computability theory , Handbook of Computability Theory (E. R. Griffor, editor), Studies in Logic and the Foundations of Mathematics, vol. 140, North-Holland, Amsterdam, 1999, pp. 3785.10.1016/S0049-237X(99)80018-4CrossRefGoogle Scholar
Cenzer, D. and Remmel, J. B., ${\varPi}_1^0$ classes in mathematics , Handbook of Recursive Mathematics , vol. 2 (Y. L. Ershov, S. S. Goncharov, A. Nerode, J. B. Remmel, and V. W. Marek, editors), Studies in Logic and the Foundations of Mathematics, vol. 139, North-Holland, Amsterdam, 1998, pages 623821.Google Scholar
Cholak, P., Coles, R., Downey, R., and Herrmann, E., Automorphisms of the lattice of ${\varPi}_1^0$ classes: Perfect thin classes and ANC degrees . Transactions of the American Mathematical Society , vol. 353 (2001), no. 12, pp. 48994924.CrossRefGoogle Scholar
Couch, P. J., Daniel, B. D., and McNicholl, T. H., Computing space-filling curves . Theory of Computing Systems , vol. 50 (2012), no. 2, pp. 370386.CrossRefGoogle Scholar
Day, A. R. and Miller, J. S., Randomness for non-computable measures . Transactions of the American Mathematical Society , vol. 365 (2013), no. 7, pp. 35753591.CrossRefGoogle Scholar
Diamondstone, D. E., Dzhafarov, D. D., and Soare, R. I., ${\varPi}_1^0$ classes, Peano arithmetic, randomness, and computable domination . Notre Dame Journal of Formal Logic , vol. 51 (2010), no. 1, pp. 127159.CrossRefGoogle Scholar
Downey, R. G., Abstract dependence, recursion theory, and the lattice of recursively enumerable filters . Bulletin of the Australian Mathematical Society , vol. 27 (1983), no. 3, pp. 461464.CrossRefGoogle Scholar
Downey, R. and Hirschfeldt, D., Algorithmic Randomness and Complexity , Theory and Applications of Computability, Springer, New York, 2010.10.1007/978-0-387-68441-3CrossRefGoogle Scholar
Downey, R. and Jockusch, C. G., Every low Boolean algebra is isomorphic to a recursive one . Proceedings of the American Mathematical Society , vol. 122 (1994), no. 3, pp. 871880.CrossRefGoogle Scholar
Downey, R. G. and Melnikov, A. G., Computable analysis and classification problems , Beyond the Horizon of Computability—16th Conference on Computability in Europe (M. Anselmo, G. D. Vedova, F. Manea, and A. Pauly, editors), Lecture Notes in Computer Science, vol. 12098, Springer, Cham, 2020, pp. 100111.CrossRefGoogle Scholar
Downey, R., Melnikov, A., and Ng, K. M., Foundations of online structure theory II: The operator approach . Logical Methods in Computer Science , vol. 17 (2021), no. 3, Paper no. 6, 35 pp.Google Scholar
Enflo, P., A counterexample to the approximation problem in Banach spaces . Acta Mathematica , vol. 130 (1973), pp. 309317.CrossRefGoogle Scholar
Ershov, Y. and Goncharov, S., Constructive Models , Siberian School of Algebra and Logic, Consultants Bureau, New York, 2000.10.1007/978-1-4615-4305-3CrossRefGoogle Scholar
Feiner, L., Hiearchies of Boolean algebras . Journal of Symbolic Logic , vol. 35 (1970), no. 3, pp. 365374.CrossRefGoogle Scholar
Franklin, J. N. Y. and McNicholl, T. H., Degrees of and lowness for isometric isomorphism . Journal of Logic and Analysis , vol. 12 (2020), Paper no. 6, 23 pp.CrossRefGoogle Scholar
Franklin, J. N. Y. and Turetsky, D., Taking the path computably traveled . Journal of Logic and Computation , vol. 29 (2019), no. 6, pp. 969973.CrossRefGoogle Scholar
Fuchs, L., Infinite Abelian Groups. Volume I , Pure and Applied Mathematics, vol. 36, Academic Press, New York, 1970.Google Scholar
Gács, P., Uniform test of algorithmic randomness over a general space . Theoretical Computer Science , vol. 341 (2005), nos. 1–3, pp. 91137.CrossRefGoogle Scholar
Gao, S., Invariant Descriptive Set Theory , Pure and Applied Mathematics, vol. 293, CRC Press, Boca Raton, 2009.Google Scholar
Gavruskin, A. and Nies, A., Universality for left-computably enumerable metric spaces . Lobachevskii Journal of Mathematics , vol. 35 (2014), no. 4, pp. 292294.CrossRefGoogle Scholar
Gherardi, G. and Marcone, A., How incomputable is the separable Hahn–Banach theorem? Notre Dame Journal of Formal Logic , vol. 50 (2010), no. 4, pp. 393425.Google Scholar
Goncharov, S., Countable Boolean Algebras and Decidability , Siberian School of Algebra and Logic, Consultants Bureau, New York, 1997.Google Scholar
Goncharov, S. and Knight, J., Computable structure and antistructure theorems . Algebra Logika , vol. 41 (2002), no. 6, pp. 639681, 757.Google Scholar
Goodstein, R., Recursive Analysis , Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1961.Google Scholar
Greenberg, N., Melnikov, A. G., Knight, J. F., and Turetsky, D., Uniform procedures in uncountable structures . Journal of Symbolic Logic , vol. 83 (2018), no. 2, pp. 529550.CrossRefGoogle Scholar
Gregoriades, V., Kispéter, T., and Pauly, A., A comparison of concepts from computable analysis and effective descriptive set theory . Mathematical Structures in Computer Science , vol. 27 (2017), no. 8, pp. 14141436.CrossRefGoogle Scholar
Grzegorczyk, A., Computable functionals . Fundamenta Mathematicae , vol. 42 (1955), pp. 168202.10.4064/fm-42-1-168-202CrossRefGoogle Scholar
Grzegorczyk, A., On the definition of computable functionals . Fundamenta Mathematicae , vol. 42 (1955), pp. 232239.CrossRefGoogle Scholar
Grzegorczyk, A., On the definitions of computable real continuous functions . Fundamenta Mathematicae , vol. 44 (1957), pp. 6171.10.4064/fm-44-1-61-71CrossRefGoogle Scholar
Harrison-Trainor, M. and Melnikov, A., An arithmetic analysis of closed surfaces. Transactions of the American Mathematical Society , to appear, https://doi.org/10.1090/tran/8915.CrossRefGoogle Scholar
Harrison-Trainor, M., Melnikov, A., and Ng, K. M., Computability of polish spaces up to homeomorphism . Journal of Symbolic Logic , vol. 85 (2020), pp. 16641686.CrossRefGoogle Scholar
Hewitt, E., The role of compactness in analysis . American Mathematical Monthly , vol. 67 (1960), no. 6, pp. 499516.Google Scholar
Hofmann, K. H. and Morris, S. A., The Structure of Compact Groups , De Gruyter Studies in Mathematics, vol. 25, De Gruyter, Berlin, 1998.Google Scholar
Hoyrup, M., Kihara, T., and Selivanov, V., Degree spectra of homeomorphism types of Polish spaces, preprint, 2020, arXiv:2004.06872.CrossRefGoogle Scholar
Hoyrup, M. and Rojas, C., Computability of probability measures and Martin–Löf randomness over metric spaces . Information and Computation , vol. 207 (2009), no. 7, pp. 830847.CrossRefGoogle Scholar
Iljazović, Z., Chainable and circularly chainable co-r.e. sets in computable metric spaces . Journal of Universal Computer Science , vol. 15 (2009), no. 6, pp. 12061235.Google Scholar
Iljazović, Z., Isometries and computability structures . Journal of Universal Computer Science , vol. 16 (2010), no. 18, pp. 25692596.Google Scholar
Iljazović, Z., Compact manifolds with computable boundaries . Logical Methods in Computer Science , vol. 9 (2013), 4, 4:19, 22 pp.Google Scholar
Iljazović, Z. and Kihara, T., Computability of subsets of metric spaces , Handbook of Computability and Complexity in Analysis (V. Brattka and P. Hertling, editors), Theory and Applications of Computability, Springer, Cham, 2021, pp. 2969.10.1007/978-3-030-59234-9_2CrossRefGoogle Scholar
Iljazović, Z. and Sušić, I., Semicomputable manifolds in computable topological spaces . Journal of Complexity , vol. 45 (2018), pp. 83114.CrossRefGoogle Scholar
Iljazović, Z. and Validžić, L., Effective compactness and orbits of points under the isometry group . Annals of Pure and Applied Logic , vol. 174 (2023), no. 2, Paper no. 103198.CrossRefGoogle Scholar
Jockusch, C. G. Jr. and Soare, R. I., A minimal pair of ${\varPi}_1^0$ classes. Journal of Symbolic Logic, vol. 36 (1971), pp. 6678.CrossRefGoogle Scholar
Jockusch, C. G. Jr. and Soare, R. I., Degrees of members of ${\varPi}_1^0$ classes. Pacific Journal of Mathematics, vol. 40 (1972), pp. 605616.10.2140/pjm.1972.40.605CrossRefGoogle Scholar
Jockusch, C. G. Jr. and Soare, R. I., ${\varPi}_1^0$ classes and degrees of theories . Transactions of the American Mathematical Society , vol. 173 (1972), pp. 3356.Google Scholar
Kalantari, I. and Weitkamp, G., Effective topological spaces. I. A definability theory . Annals of Pure and Applied Logic , vol. 29 (1985), no. 1, pp. 127.CrossRefGoogle Scholar
Kalantari, I. and Weitkamp, G., Effective topological spaces. II. A hierarchy . Annals of Pure and Applied Logic , vol. 29 (1985), no. 2, pp. 207224.CrossRefGoogle Scholar
Kalantari, I. and Weitkamp, G., Effective topological spaces. III. Forcing and definability . Annals of Pure and Applied Logic , vol. 36 (1987), no. 1, pp. 1727.CrossRefGoogle Scholar
Kalantari, I. and Welch, L., On Turing degrees of points in computable topology . Mathematical Logic Quarterly , vol. 54 (2008), no. 5, pp. 470482.CrossRefGoogle Scholar
Kamo, H., Computability in some fundamental theorems in functional analysis and general topology , Ph.D. thesis, Kyoto University, 2006.Google Scholar
Khisamiev, N., Hierarchies of torsion-free abelian groups . Algebra Logika , vol. 25 (1986), no. 2, pp. 205226, 244.CrossRefGoogle Scholar
Kleene, S. C., A note on computable functionals . Nederlandse Akademie van Wetenschappen. Proceedings. Series A , vol. 18 (1956), pp. 275280.Google Scholar
Knight, J. F. and Stob, M., Computable Boolean algebras . Journal of Symbolic Logic , vol. 65 (2000), no. 4, pp. 16051623.CrossRefGoogle Scholar
Ko, K.-I., Complexity Theory of Real Functions , Progress in Theoretical Computer Science, Birkhäuser, Boston, 1991.CrossRefGoogle Scholar
Ko, K.-I. and Friedman, H., Computational complexity of real functions . Theoretical Computer Science , vol. 20 (1982), no. 3, pp. 323352.CrossRefGoogle Scholar
Korovina, M. and Kudinov, O., Towards computability over effectively enumerable topological spaces , Proceedings of the Fifth International Conference on Computability and Complexity in Analysis (V. Brattka, R. Dillhage, T. Grubba, and A. Klutsch, editors), Electronic Notes in Theoretical Computer Science, vol. 221, Elsevier, Amsterdam, 2008, pp. 115125.Google Scholar
Korovina, M. and Kudinov, O., The Rice–Shapiro theorem in computable topology . Logical Methods in Computer Science , vol. 13 (2017), no. 4, Paper no. 30, 13 pp.Google Scholar
Korovina, M. and Kudinov, O., Highlights of the Rice–Shapiro theorem in computable topology , Perspectives of System Informatics (A. Pnueli, I. Virbitskaite, and A. Voronkov, editors), Lecture Notes in Computer Science, vol. 10742, Springer, Cham, 2018, pp. 241255.CrossRefGoogle Scholar
Kreisel, G., A variant to Hilbert’s theory of the foundations of arithmetic . British Journal for the Philosophy of Science , vol. 4 (1953), pp. 107129; errata and corrigenda, vol. 357 (1954).CrossRefGoogle Scholar
Kudinov, O. and Selivanov, V., First order theories of some lattices of open sets . Logical Methods in Computer Science , vol. 13 (2017), no. 3, Paper no. 16, 18 pp.Google Scholar
Lacombe, D., Quelques procédés de définition en topologie recursive , Constructivity in Mathematics: Proceedings of the Colloquium Held at Amsterdam, 1957 (A. Heyting, editor), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1959, pp. 129158.Google Scholar
Lešnik, D., Constructive Urysohn universal metric space . Journal of Universal Computer Science , vol. 15 (2009), no. 6, pp. 12361263.Google Scholar
Long, Q., Computability-theoretic complexity of effective Banach spaces , M.Sc. thesis, Victoria University of Wellington, 2021.Google Scholar
Lupini, M., Melnikov, A., and Nies, A., Computable topological abelian groups . Journal of Algebra , vol. 615 (2023), pp. 278327.10.1016/j.jalgebra.2022.10.003CrossRefGoogle Scholar
Marcone, A. and Valenti, M., Effective aspects of Hausdorff and Fourier dimension . Computability , vol. 11 (2022), nos. 3–4, pp. 299333.CrossRefGoogle Scholar
Melnikov, A., Enumerations and completely decomposable torsion-free abelian groups . Theory of Computing Systems , vol. 45 (2009), no. 4, pp. 897916.CrossRefGoogle Scholar
Melnikov, A. G., Computably isometric spaces . Journal of Symbolic Logic , vol. 78 (2013), no. 4, pp. 10551085.CrossRefGoogle Scholar
Melnikov, A., Computable topological groups and Pontryagin duality . Transactions of the American Mathematical Society , vol. 370 (2018), no. 12, pp. 87098737.CrossRefGoogle Scholar
Melnikov, A., New degree spectra of polish spaces . Rossiyskaya Akademiya Nauk , vol. 62 (2021), no. 5, pp. 10911108.Google Scholar
Melnikov, A. and Montalbán, A., Computable polish group actions . Journal of Symbolic Logic , vol. 83 (2018), no. 2, pp. 443460.10.1017/jsl.2017.68CrossRefGoogle Scholar
Melnikov, A. G. and Ng, K. M., Computable structures and operations on the space of continuous functions . Fundamenta Mathematicae , vol. 233 (2016), no. 2, pp. 101141.Google Scholar
Melnikov, A. G. and Nies, A., The classification problem for compact computable metric spaces , The Nature of Computation (P. Bonizzoni, V. Brattka, and B. Löwe, editors), Lecture Notes in Computer Science, vol. 7921, Springer, Heidelberg, 2013, pp. 320328.Google Scholar
Metakides, G. and Nerode, A., Recursively enumerable vector spaces . Annals of Mathematical Logic , vol. 11 (1977), no. 2, pp. 147171.CrossRefGoogle Scholar
Metakides, G. and Nerode, A., Effective content of field theory . Annals of Mathematical Logic , vol. 17 (1979), no. 3, pp. 289320.CrossRefGoogle Scholar
Metakides, G., Nerode, A., and Shore, R. A., Recursive limits on the Hahn–Banach theorem , Errett Bishop: Reflections on Him and His Research (M. Rosenbla, editor), Contemporary Mathematics, vol. 39, American Mathematical Society, Providence, 1985, pp. 8591.CrossRefGoogle Scholar
Miller, J. S., $\boldsymbol{\varPi}_{\boldsymbol{1}}^{\boldsymbol{0}}$ classes in computable analysis and topology , Ph.D. thesis, Cornell University, Ithaca, 2002.Google Scholar
Mori, T., Tsujii, Y., and Yasugi, M., Computability structures on metric spaces , Combinatorics, Complexity, and Logic (D. S. Bridges, C. S. Calude, J. Gibbons, S. Reeves, and I. H. Witten, editors), Discrete Mathematics and Theoretical Computer Science, vol. 1, Springer, Singapore, 1997, pp. 351362.Google Scholar
Moschovakis, Y. N., Recursive metric spaces . Fundamenta Mathematicae , vol. 55 (1964), pp. 215238.CrossRefGoogle Scholar
Moschovakis, Y. N., Descriptive Set Theory , second ed., Mathematical Surveys and Monographs, vol. 155, American Mathematical Society, Providence, 2009.CrossRefGoogle Scholar
Mummert, C., Reverse mathematics of MF spaces . Journal of Mathematical Logic , vol. 6 (2006), no. 2, pp. 203232.CrossRefGoogle Scholar
Munkres, J. R., Elements of Algebraic Topology , Addison-Wesley, Menlo Park, 1984.Google Scholar
Myhill, J., A recursive function, defined on a compact interval and having a continuous derivative that is not recursive . Michigan Mathematical Journal , vol. 18 (1971), no. 2, pp. 9798.CrossRefGoogle Scholar
Nerode, A. and Huang, W. Q., An application of pure recursion theory to recursive analysis . Acta Mathematica Sinica , vol. 28 (1985), no. 5, pp. 625636.Google Scholar
Neumann, E., On the computability of the set of automorphisms of the unit square . Theoretical Computer Science , vol. 903 (2022), pp. 7483.CrossRefGoogle Scholar
Nies, A. (editor), Logic Blog 2016, preprint available from arXiv:1703.01573.Google Scholar
Nies, A. and Solecki, S., Local compactness for computable Polish metric spaces is ${\varPi}_1^1$ -complete, Evolving Computability (A. Beckmann, V. Mitrana, and M. Soskova, editors), Proceedings of 11th Conference on Computability in Europe, CiE 2015, Bucharest, Romania, June 29-July 3, 2015, pp. 286290. Available at https://link.springer.com/book/10.1007/978-3-319-20028-6 CrossRefGoogle Scholar
Nogina, E. J., Effectively topological spaces . Doklady Akademii Nauk SSSR , vol. 169 (1966), pp. 2831.Google Scholar
Nogina, E. J., Correlations between certain classes of effectively topological spaces . Matematicheskie Zametki , vol. 5 (1969), pp. 483495.Google Scholar
Odintsov, S. P. and Selivanov, V. L., Arithmetic hierarchy and ideals of enumerated Boolean algebras . Siberian Mathematical Journal , vol. 30 (1989), no. 6, pp. 952960.CrossRefGoogle Scholar
Pauly, A., On the topological aspects of the theory of represented spaces . Computability , vol. 5 (2016), no. 2, pp. 159180.CrossRefGoogle Scholar
Pauly, A., Enumeration degrees and topology , Sailing Routes in the World of Computation (F. Manea, R. G. Miller, and D. Nowotka, editors), Lecture Notes in Computer Science, vol. 10936, Springer, Cham, 2018, pp. 328337.CrossRefGoogle Scholar
Pauly, A., Effective local compactness and the hyperspace of located sets, preprint, 2019, arXiv:1903.05490.Google Scholar
Pauly, A., Seon, D., and Ziegler, M., Computing Haar measures , 28th EACSL Annual Conference on Computer Science Logic (M. Fernández and A. Muscholl, editors), Leibniz International Proceedings in Informatics (LIPIcs), vol. 152, Schloss Dagstuhl—Leibniz-Zentrum fuer Informatik, Wadern, 2020, pp. 4:134:17.Google Scholar
Pontryagin, L. S., Foundations of Algebraic Topology , third ed., Nauka, Moscow, 1986.Google Scholar
Pour-El, M. B. and Caldwell, J., On a simple definition of computable function of a real variable—With applications to functions of a complex variable . Mathematical Logic Quarterly , vol. 21 (1975), no. 1, pp. 119.CrossRefGoogle Scholar
Pour-El, M. B. and Richards, I., Computability and noncomputability in classical analysis . Transactions of the American Mathematical Society , vol. 275 (1983), no. 2, pp. 539560.CrossRefGoogle Scholar
Pour-El, M. B. and Richards, J. I., Computability in Analysis and Physics , Perspectives in Mathematical Logic, Springer, Berlin, 1989.CrossRefGoogle Scholar
Reimann, J. and Slaman, T. A., Measures and their random reals . Transactions of the American Mathematical Society , vol. 367 (2015), no. 7, pp. 50815097.CrossRefGoogle Scholar
Rettinger, R., Compactness and the effectivity of uniformization , How the World Computes (S. B. Cooper, A. Dawar, and B. Löwe, editors), Lecture Notes in Computer Science, vol. 7318, Springer, Heidelberg, 2012, pp. 616625.CrossRefGoogle Scholar
La Roche, P. E., Contributions to recursive algebra , Ph.D. thesis, Cornell University, ProQuest LLC, Ann Arbor, 1978.Google Scholar
La Roche, P., Effective Galois theory . Journal of Symbolic Logic , vol. 46 (1981), no. 2, pp. 385392.CrossRefGoogle Scholar
Rogers, H.. Theory of Recursive Functions and Effective Computability , second ed., MIT Press, Cambridge, 1987.Google Scholar
Sagan, H., Space-Filling Curves , Universitext, Springer, New York, 1994.CrossRefGoogle Scholar
Selivanov, V. and Selivanova, S., Primitive recursive ordered fields and some applications. Computability , vol. 12 (2023), no. 1, pp. 7199.Google Scholar
Selivanov, V. L. and Selivanova, S., Primitive recursive ordered fields and some applications , Computer Algebra in Scientific Computing (F. Boulier, M. England, T. M. Sadykov, and E. V. Vorozhtsov, editors), Lecture Notes in Computer Science, vol. 12865, Springer, Cham, 2021, pp. 353369.CrossRefGoogle Scholar
Shoenfield, J. R., Degrees of models . Journal of Symbolic Logic , vol. 25 (1960), pp. 233237.CrossRefGoogle Scholar
Sierpiński, W., Sur un espace métrique séparable universel . Fundamenta Mathematicae , vol. 33 (1945), pp. 115122.CrossRefGoogle Scholar
Simpson, S., Subsystems of Second Order Arithmetic , second ed., Perspectives in Logic, Cambridge University Press, Cambridge, 2009.CrossRefGoogle Scholar
Smith, R. L., The theory of profinite groups with effective presentations , Ph.D. thesis, The Pennsylvania State University, ProQuest LLC, Ann Arbor, 1979.Google Scholar
Smith, R. L., Effective aspects of profinite groups . Journal of Symbolic Logic , vol. 46 (1981), no. 4, pp. 851863.CrossRefGoogle Scholar
Specker, E., Der Satz vom Maximum in der rekursiven Analysis , Constructivity in Mathematics: Proceedings of the Colloquium Held at Amsterdam, 1957 (A. Heyting, editor), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1959, pp. 254265.Google Scholar
Spreen, D., A characterization of effective topological spaces , Recursion Theory Week (Oberwolfach, 1989) (K. Ambos-Spies, G. H. Müller, and G. E. Sacks, editors), Lecture Notes in Mathematics, vol. 1432, Springer, Berlin, 1990, pp. 363387.CrossRefGoogle Scholar
Spreen, D., Representations versus numberings: On the relationship of two computability notions . Theoretical Computer Science , vol. 262 (2001), nos. 1–2, pp. 473499.CrossRefGoogle Scholar
Spreen, D. and Young, P., Effective operators in a topological setting , Computation and Proof Theory (Aachen, 1983) (E. Börger, W. Oberschelp, M. M. Richter, B. Schinzel, and W. Thomas, editors), Lecture Notes in Mathematics, vol. 1104, Springer, Berlin, 1984, pp. 437451.CrossRefGoogle Scholar
Tran, Y.-Y., Computably enumerable Boolean algebras . Ph.D. thesis, Cornell, 2018. Available at https://ecommons.cornell.edu/handle/1813/59504 Google Scholar
Turing, A. M., On computable numbers, with an application to the Entscheidungsproblem . Proceedings of the London Mathematical Society , vol. 42 (1936), pp. 230265.Google Scholar
Turing, A. M., On computable numbers, with an application to the Entscheidungsproblem. A correction. Proceedings of the London Mathematical Society , vol. 43 (1937), pp. 544546.Google Scholar
Waterhouse, W. C., Profinite groups are Galois groups . Proceedings of the American Mathematical Society , vol. 42 (1974), pp. 639640.Google Scholar
Weihrauch, K., Computable Analysis: An Introduction , Texts in Theoretical Computer Science. An EATCS Series, Springer, Berlin, 2000.CrossRefGoogle Scholar
Weihrauch, K., Computational complexity on computable metric spaces . Mathematical Logic Quarterly , vol. 49 (2003), no. 1, pp. 321.CrossRefGoogle Scholar
Weihrauch, K. and Grubba, T., Elementary computable topology . Journal of Universal Computer Science , vol. 15 (2009), no. 6, pp. 13811422.Google Scholar
Zaslavskiĭ, I. D., Some properties of constructive real numbers and constructive functions . Trudy Matematicheskogo Instituta Imeni V. A. Steklova , vol. 67 (1962), pp. 385457.Google Scholar