Bauer, Andrej (2000). *The Realizability Approach to Computable Analysis and Topology*. PhD thesis, Carnegie Mellon University. Technical report CMU-CS-00-164.

Bauer, Andrej (2008). Efficient computation with Dedekind reals. In Fifth International Conference on Computability and Complexity in Analysis, Vasco, Brattka, Ruth, Dillhage, Tanja, Grubba, and Klutch, Angela, editors, Hagen, Germany.

Bauer, Andrej and Kavkler, Iztok (2008). Implementing real numbers with RZ. Electronic Notes in Theoretical Computer Science, 202:365–384.

Bauer, Andrej and Stone, Christopher (2008). RZ: a tool for bringing constructive and computable mathematics closer to programming practice. Journal of Logic and Computation, 19:17–43.

Berger, Josef and Bridges, Douglas (2008a). The anti-Specker property, a Heine–Borel property, and uniform continuity. Archive for Mathematical Logic, 46:583–592.

Berger, Josef and Bridges, Douglas (2008b). The fan theorem and positive-valued uniformly continuous functions on compact intervals. New Zealand Journal of Mathematics, 38:129–135.

Bishop, Errett (1967). Foundations of Constructive Analysis. Higher Mathematics. McGraw–Hill.

Bishop, Errett and Bridges, Douglas (1985). Constructive Analysis. Number 279 in Grundlehren der mathematischen Wissenschaften. Springer-Verlag.

Bourbaki, Nicolas (1966). Topologie Générale. Hermann. English translation, *General Topology*, distributed by Springer-Verlag, 1989.

Bridges, Douglas (1999). Constructive mathematics: a foundation for computable analysis. Theoretical Computer Science, 219:95–109.

Bridges, Douglas and Richman, Fred (1987). Varieties of Constructive Mathematics. Number 97 in London Mathematical Society Lecture Notes. Cambridge University Press.

Cederquist, Jan and Negri, Sara (1996). A constructive proof of the Heine–Borel covering theorem for formal reals. In Types for Proofs and Programs, Stefano, Beradi and Mario, Coppo, editors, number 1158 in *Lecture Notes in Computer Science*. Springer-Verlag.

Cleary, John (1987). Logical arithmetic. Future Computing Systems, 2 (2):125–149.

Conway, John Horton (1976). On Numbers and Games. Number 6 in London Mathematical Society Monographs. Academic Press. Revised edition, 2001, published by A K Peters, Ltd.

Dedekind, Richard (1872). *Stetigkeit und irrationale Zahlen*. Braunschweig. Reprinted in (Dedekind 1932), pages 315–334; English translation, *Continuity and Irrational Numbers*, in (Dedekind 1901).

Dedekind, Richard (1901). Essays on the theory of numbers. Open Court. English translations by Beman, Wooster Woodruff; republished by Dover, 1963.

Dedekind, Richard (1932). Gesammelte mathematische Werke, volume 3. Vieweg, Braunschweig. Edited by Fricke, Robert, Noether, Emmy and Ore, Øystein; republished by Chelsea, New York, 1969.

Edalat, Abbas and Sünderhauf, Philipp (1998). A domain-theoretic approach to real number computation. Theoretical Computer Science, 210:73–98.

Escardó, Martín (2004). Synthetic topology of data types and classical spaces. Electronic Notes in Theoretical Computer Science, 87:21–156.

Fourman, Michael and Hyland, Martin (1979). Sheaf models for analysis. In Applications of Sheaves, Michael, Fourman, Chris, Mulvey, and Scott, Dana, editors, number 753 in Lecture Notes in Mathematics. Springer-Verlag.

Friedberg, Richard (1958). Un contre-example relatif aux fonctionnelles récursives. Comptes rendus hebdomadaires des scéances de l'Académie des Sciences (Paris), 247:852–854.

Gentzen, Gerhard (1935). Untersuchungen über das logische Schliessen. Mathematische Zeitschrift, 39:176–210, 405–431. English translation in (Gentzen 1969), pages 68–131.

Gentzen, Gerhard (1969). The Collected Papers of Gerhard Gentzen. Studies in Logic and the Foundations of Mathematics. North-Holland. Edited by Manfred, Szabo.

Gierz, Gerhard, Hoffmann, Karl Heinrich, Keimel, Klaus, Lawson, Jimmie, Mislove, Michael, and Scott, Dana (1980). A Compendium of Continuous Lattices. Springer-Verlag. Second edition, *Continuous Lattices and Domains*, published by Cambridge University Press, 2003.

Hanrot, Guillaume, Lefèvre, Vincent, Pélissier, Patrick, and Zimmermann, Paul. *The MPFR Library*. INRIA. www.mpfr.org. Heyting, Arend (1956). Intuitionism, an Introduction. Studies in Logic and the Foundations of Mathematics. North-Holland. Third edition, 1971.

Hyland, Martin (1982). The effective topos. In The L.E.J. Brouwer Centenary Symposium, Anne, Troelstra and van Dalen, Dirk, editors. North Holland.

Hyland, Martin (1991). First steps in synthetic domain theory. In Category Theory, Aurelio, Carboni, Maria-Cristina, Pedicchio, and Rosolini, Giuseppe, editors, number 1488 in Lecture Notes in Mathematics. Springer Verlag.

Hyland, Martin and Rosolini, Giuseppe (1990). The discrete objects in the effective topos. Proceedings of the London Mathematical Society, 60:1–36.

Johnstone, Peter (1977). Topos Theory. Number 10 in London Mathematical Society Monographs. Academic Press.

Johnstone, Peter (1982). Stone Spaces. Number 3 in Cambridge Studies in Advanced Mathematics. Cambridge University Press.

Johnstone, Peter (1984). Open locales and exponentiation. Contemporary Mathematics, 30:84–116.

Joyal, André and Tierney, Myles (1984). An Extension of the Galois Theory of Grothendieck, volume 309 of *Memoirs*. American Mathematical Society.

Julian, William and Richman, Fred (1984). A uniformly continuous function on [0,1] that is everywhere different from its infimum. Pacific Journal of Mathematics, 111:333–340.

Kaucher, Edgar (1980). Interval analysis in the extended interval space *IR*. In Fundamentals of Numerical Computation, Götz, Alefeld and Grigorieff, Rolf, editors, volume 2 of *Computing. Supplementum*. Springer-Verlag.

Kearfott, Baker (1996). Interval computations: Introduction, uses and resources. Euromath Bulletin, 2 (1):95–112.

Kelly, Max (1986). A survey of totality in ordinary and enriched categories. Cahiers de Géometrie et Topologie Differentielle, 27:109–132.

Kleene, Stephen (1945). On the interpretation of intuitionistic number theory. Journal of Symbolic Logic, 10:109–124.

Kock, Anders (2006). Synthetic Differential Geometry. Number 333 in London Mathematical Society Lecture Note Series. Cambridge University Press, second edition.

Kreinovich, Vladik, Nesterov, Vyacheslav, and Zheludeva, Nina (1996). Interval methods that are guaranteed to underestimate (and the resulting new justification of Kaucher arithmetic). Reliable Computing, 2 (2):119–124.

Lakeyev, Anatoly (1995). Linear algebraic equations in Kaucher arithmetic. In *Applications of Interval Computations (APIC'95)*, Vladik, Kreinovich, editor. supplement to *Reliable Computing*.

Lambov, Branimir (2007). RealLib: An efficient implementation of exact real arithmetic. Mathematical Structures in Computer Science, 17 (1):81–98. www.brics.dk/~barnie/RealLib/. Moore, Ramon (1966). Interval Analysis. Prentice Hall.

Müller, Norbert (2001). The iRRAM: Exact arithmetic in C++. In Computability and Complexity in Analysis: 4th International Workshop, CCA 2000 Swansea, UK, September 17, 2000, Selected Papers, Jens, Blanck, Vasco, Brattka, and Hertling, Peter, editors, number 2064 in Lecture Notes in Computer Science. Springer-Verlag.

Palmgren, Eric (2005). Continuity on the real line and in formal spaces. In From Sets and Types to Topology and Analysis: Towards Practicable Foundations of Constructive Mathematics, Laura, Crosilla and Schuster, Peter, editors, Oxford Logic Guides. Oxford University Press.

Peano, Giuseppe (1897). Studii di logica matematica. Atti della Reale Accademia di Torino, 32:565–583. Reprinted in Peano, *Opere Scelte*, Cremonese, 1953, vol. 2, pp. 201–217, and (in English) in Kennedy, Hubert, *Selected Works of Giuseppe Peano*, Toronto University Press, 1973, pp 190–205.

Plotkin, Gordon (1977). LCF considered as a programming language. Theoretical Computer Science, 5:223–255.

Robinson, Abraham (1966). Non-standard Analysis. North-Holland. Revised edition, 1996, published by Princeton University Press.

Rogers, Hartley (1992). Theory of Recursive Functions and Effective Computability. MIT Press, third edition.

Rosemeier, Frank (2001). A constructive approach to Conway's theory of games. In Reuniting the Antipodes: Constructive and Nonstandard Views of the Continuum, Schuster, Peter, Berger, Ulrich, and Osswald, Horst, editors. Springer-Verlag.

Rosolini, Giuseppe (1986). *Continuity and Effectiveness in Topoi*. PhD thesis, University of Oxford.

Schuster, Peter (2003). Unique existence, approximate solutions, and countable choice. Theoretical Computer Science, 305:433–455.

Schuster, Peter (2005). What is continuity, constructively? Journal of Universal Computer Science, 11 (12):2076–2085.

Scott, Dana (1972a). Continuous lattices. In Toposes, Algebraic Geometry and Logic, Bill, Lawvere, editor, number 274 in Lecture Notes in Mathematics. Springer-Verlag.

Scott, Dana (1972b). Lattice theory, data types and semantics. In Formal Semantics of Programming Languages, Rustin, Randall, editor. Prentice-Hall.

Spitters, Bas (2007). Located and overt sublocales. arxiv.org/abs/math/0703561.

Stevenson, David (1985). Binary floating-point arithmetic. ANSI/IEEE Standard, 754. Revised, 2008.

Stolz, Otto (1883). Zur Geometrie der Alten, insbesondere über ein Axiom des Archimedes. Mathematische Annalen, 22 (4):504–519.

Street, Ross (1980). Cosmoi of internal categories. Transactions of the American Mathematical Society, 258:271–318.

Street, Ross and Walters, Robert (1978). Yoneda structures on 2-categories. Journal of Algebra, 50:350–379.

Taylor, Paul (1991). The fixed point property in synthetic domain theory. In Logic in Computer Science 6, Kahn, Gilles, editor. IEEE.

Taylor, Paul (1999). Practical Foundations of Mathematics. Number 59 in Cambridge Studies in Advanced Mathematics. Cambridge University Press.

[O]Taylor, Paul (2009) Foundations for Computable Topology. In Foundational Theories of Mathematics, Sommaruga, Giovanni, editor, Kluwer.

[A]Taylor, Paul (2002) Sober spaces and continuations. Theory and Applications of Categories, 10:248–299.

[B]Taylor, Paul (2002) Subspaces in abstract Stone duality. Theory and Applications of Categories, 10:300–366.

[C]Taylor, Paul (2000) Geometric and higher order logic using abstract Stone duality. Theory and Applications of Categories, 7:284–338.

[D]Taylor, Paul (2000) Non-Artin gluing in recursion theory and lifting in abstract Stone duality.

[E]Taylor, Paul (2005) Inside every model of Abstract Stone Duality lies an Arithmetic Universe. Electronic Notes in Theoretical Computer Science 122:247–296, Elsevier.

[F]Taylor, Paul (2002) Scott domains in abstract Stone duality.

[G]Taylor, Paul (2006) Computably based locally compact spaces. Logical Methods in Computer Science, 2:1–70.

[H]Taylor, Paul (2004) An elementary theory of various categories of spaces and locales.

[J]Taylor, Paul (2005) A lambda calculus for real analysis. In *Computability and Complexity in Analysis*, Tanja, Grubba et al. , editors, volume 326 of *Informatik Berichte*, FernUniversität in Hagen.

[K]Taylor, Paul (2006) Interval analysis without intervals. In *Real Numbers and Computers*, Guillaume, Hanrot and Zimmermann, Paul, editors, Nancy.

[L]Taylor, Paul (2004) Tychonov's theorem in abstract Stone duality.

[M]Taylor, Paul (2009) Cartesian closed categories with subspaces.

Thielecke, Hayo (1997). *Categorical Structure of Continuation Passing Style*. PhD thesis, University of Edinburgh. ECS-LFCS-97-376.

Tholen, Walter (1980). A note on total categories. Bulletin of the Australian Mathematical Society, 21:169–173.

Troelstra, Anne Sjerp and van Dalen, Dirk (1988). Constructivism in Mathematics, an Introduction. Numbers 121 and 123 in Studies in Logic and the Foundations of Mathematics. North-Holland.

Vermeulen, Japie (1994). Proper maps of locales. Journal of Pure and Applied Algebra, 92:79–107.

Waaldijk, Frank (2005). On the foundations of constructive mathematics – especially in relation to the theory of continuous functions. Foundations of Science, 10 (3):249–324.

Weihrauch, Klaus (2000). Computable Analysis. Springer-Verlag, Berlin.

Wood, Richard (1982). Some remarks on total categories. Journal of Algebra, 75:538–545.