POSTLUDE: PATHS TO TOPOLOGY

There are nine and sixty ways of constructing tribal lays, And every single one of them is right! Rudyard Kipling

Other than the one we have followed in the foregoing chapters, the main constructive approaches to topology include: – a theory of neighbourhood spaces; – the theory of frames/locales; – formal topology. The first of these, originally suggested by Bishop [9, 12], has been connected by Ishihara et al. [48, 59] to the theory of quasi-apartness spaces—essentially spaces with a relation ./ that satisfy axioms A1–A4 but with the complement replaced by the logical complement in A2 and A4. In particular, the paper [59] shows how to construct an adjunction between the category1 of neighbourhood spaces and that of quasi-apartness spaces. Both the theory of frames/locales and formal topology produce point-free topology: topology in which the notion of point is secondary to that cor- responding to open set. In the former case, the fundamental structure is a frame, defined to be a complete lattice L in which finite meets distribute over arbitrary unions: that is, ! _ _ x ∧ yi = (x ∧ yi) i∈I i∈I

1In this postlude we assume some familiarity with categories, lattice theory, and orderings.

D.S. Bridges and L.S. Vîţă, Apartness and Uniformity: A Constructive Development, 183 Theory and Applications of Computability, DOI 10.1007/978-3-642-22415-7, © Springer-Verlag Berlin Heidelberg 2011 184 Postlude: Paths to Topology

holds for all x ∈ L and all families (yi)i∈I of elements of L. Taken with supremum-preserving lattice homomorphisms, frames form a category that generalises the category of topological spaces with continuous functions as morphisms. Since in the latter category it is not the image, but the inverse image, of an open set that is open, it makes sense to consider the dual of the former category; that dual is called the category of locales. We shall not discuss locales further here,2 since the resulting development of topology, though constructive, is impredicative and there is a purely predicative constructive theory—formal topology—that we now discuss briefly. This was introduced in the mid 1980s by Sambin and Martin-L¨of[79, 80, 81], in order to provide a theory of topology that was based on the latter’s type theory [69, 70, 71], which is one of the main alternatives proposed as a foundation for BISH. The fundamental notions of formal topology are

– a preordered set (A, 6) whose elements are called basic neighbourhoods, and

– a so-called covering relation C between elements of A and subsets of A, satisfying four simple axioms. Although points do not have the status in for- mal topology that they have in classical topology or our theory of apartness spaces, one can introduce a notion of point as a special type of subset α of A; roughly, a point is identified with the set of all its neighbourhoods. Denoting the collection3 of points by Pt(A), to each set U of basic neighbourhoods we associate a collection U ∗ of points as follows:

∗ U ≡ {α ∈ Pt(A): ∃a∈U (a ∈ α)} . It turns out that these collections form a topology, in the usual sense, on Pt(A). Moreover, when we look at the so-called continuous morphisms, or 0 0 0 approximable mappings, between formal topologies (A, 6, C) and (A , 6 , C ), we see that to each such relation F ⊂ A×B there is associated a point function Pt(F ) : Pt(A) → Pt(A0) defined by

0 0 0 Pt(F )(α) ≡ {a ∈ A : ∃a∈α (aF a )} . An important feature of formal topology (and of locale theory) is a notion of compactness analogous to compactness as defined, both classically and intuitionistically, in terms of open covers. By using formal covers, rather than pointwise ones, the formal topologist avoids the need for Brouwer’s fan theorem or its classical counterpart, K¨onig’slemma, when establishing, for example, the compactness of the interval [0, 1].

2For more on topology via locales, see the books by Johnstone [60] and Vickers [89]. 3We use the word ‘collection’ since in a predicative foundation like Martin-L¨oftype theory, Pt(A) may not be a set. Postlude: Paths to Topology 185

How does apartness arise within the context of a formal topology A? We say that two basic neighbourhoods a, b in A are apart, and we write a ⊥ b, if a ∧ b C ∅. This gives rise to notions of – inequality/apartness between points α, β: α 6= β if and only if a ⊥ b for some a ∈ α and b ∈ β;

– apartness between a point α and a subset S of A: α ./ S if and only if there exists a ∈ α such that for each β ∈ S, there exists β ∈ β with a ⊥ b.

It transpires that if the formal topology satisfies a certain regularity prop- erty, then X ≡ Pt(A), taken with the foregoing point-point inequality and apartness between points and sets, satisfies our axioms A1–A5 for an apart- ness space. Moreover, if F : A → A0 is an approximable mapping between formal topologies, then the associated mapping Pt(F ) : Pt(A) → Pt(A0) is continuous in the point-set apartness sense introduced on page 36. To extend these ideas to apartness between subsets of our formal space A, first define apartness of two sets U, V of basic neighbourhoods by

U ⊥ V ⇔ ∀u∈U ∀v∈V (u ⊥ v) .

For subsets S, T of A, We then define the apartness between subsets S, T of A by ∗ ∗ S ./ T ⇔ ∃U,V ⊂A (S ⊂ U , T ⊂ V , and U ⊥ V ) . When A is regular as a formal topology, the relation satisfies our axioms B1, B3–B5 (and hence B4) and this weak form of B2:

S ./ T ⇒ S ∩ T = ∅. (In fact, we do not need the regularity property of A to establish B1, the weak form of B2, and B3–B4.) Furthermore, if F : A → A0 is an approximable mapping between formal topologies, then Pt(F ) : Pt(A) → Pt(A0) is strongly continuous. Thus there is a connection from formal topologies to apartness spaces (or at least, in the set-set case, spaces which differ from apartness ones by having a weaker form of B2). For the reverse connection, we need only pass from the apartness topology associated with a given apartness space (X, ./) to the corresponding formal topology. While on the subject of lattice-based approaches to topology, we would point out that our theory of apartness between sets has been abstracted to one of apartness on frames [17, 18, 31]. REFERENCES

[1] O. Aberth: Computable Analysis, McGraw-Hill, New York, 1980.

[2] P. Aczel and M. Rathjen: Notes on Constructive Set Theory, Report No. 40, Institut Mittag-Leffler, Royal Swedish Academy of Sciences, 2001.

[3] P. Aczel and M. Rathjen: Constructive Set Theory, in preparation.

[4] B. Banaschewski, T. Coquand, and G. Sambin (eds): Second Workshop on Formal Topology (Venice, 2002), Ann. Pure Appl. Logic 137, 1–3 (special issue), 2006.

[5] A. Bauer: ‘Realizability as the Connection between Computable and Con- structive Mathematics’, preprint, University of Ljubljana, Slovenia, 2005.

[6] M.J. Beeson: Foundations of Constructive Mathematics, Springer Verlag, Heidelberg, Germany, 1985.

[7] H.L. Bentley, H. Herrlich, and M. Huˇsek:‘The historical development of uniform, proximal, and nearness concepts in topology’, in Handbook of the History of General Topology, Vol. 2 (C.E. Aull and R. Lowen, eds), 577–629, Kluwer, Amsterdam, The Netherlands, 1998.

[8] J. Berger, H. Ishihara, E. Palmgren, and P.M. Schuster: ‘A predicative completion of a uniform space’, preprint, University of Munich, Germany, 2010.

[9] E.A. Bishop: Foundations of Constructive Analysis, McGraw-Hill, New York, 1967.

[10] E.A. Bishop: ‘Schizophrenia in Contemporary Mathematics’, in Errett Bishop, Reflections on Him and His Research, Contemp. Math. 39, 1– 32, Am. Math. Soc., Providence RI, 1985.

187 188 References

[11] E.A. Bishop: The Neat Category of Stratified Spaces, preprint, University of California, San Diego, 1971.

[12] E.A. Bishop and D.S. Bridges: Constructive Analysis, Grundlehren der Math. Wissenschaften 279, Springer Verlag, Heidelberg-Berlin-New York, 1985.

[13] N. Bourbaki: General Topology (2 vols: Parts 1 and 2), Addison-Wesley, Reading, Mass., 1966.

[14] D.S. Bridges: ‘A constructive Morse theory of sets’, in Mathematical Logic and its Applications (D. Skordev, ed.), Plenum, New York, 61–79, 1987.

[15] D.S. Bridges: ‘A constructive look at the real number line’, in Synthese: Real Numbers, Generalizations of the Reals and Theories of Continua (P. Ehrlich, ed.), 29–92, Kluwer Academic, Amsterdam, The Netherlands, 1994.

[16] D.S. Bridges: ‘Constructive mathematics: a foundation for computable analysis’, Theor. Comp. Sci. 219(1-2), 95–109, 1999.

[17] D.S. Bridges: ‘Product a-frames and proximity’, Math. Logic Q. 54(1), 12–25, 2008.

[18] D.S. Bridges: ‘Almost new pre-apartness from old’, Ann. Pure Appl. Logic, to appear.

[19] D.S. Bridges: ‘Precompact apartness spaces’, preprint, University of Can- terbury, New Zealand, 2011.

[20] D.S. Bridges: ‘Some new compactness notions for an apartness space’, preprint, University of Canterbury, New Zealand, 2011.

[21] D.S. Bridges and H. Diener: ‘A constructive treatment of Urysohn’s lemma in an apartness space’, Math. Logic Q. 52(5), 464–469, 2006.

[22] D.S. Bridges, H. Ishihara, R. Mines, F. Richman, P.M. Schuster, and L.S. Vˆıt¸˘a: ‘Almost locatedness in uniform spaces’, Czech. Math. J. 57(1), 1– 12, 2007.

[23] D.S. Bridges, H. Ishihara, P.M. Schuster, and L.S. Vˆıt¸˘a:‘Strong continuity implies uniform sequential continuity’, Arch. Math. Logic 44(7), 887–895, 2005.

[24] D.S. Bridges and F. Richman: Varieties of Constructive Mathematics, London Math. Soc. Lecture Notes 97, Cambridge Univ. Press, Cambridge, U.K., 1987. References 189

[25] D.S. Bridges, P.M. Schuster, and L.S. Vˆıt¸˘a: ‘Apartness as a relation be- tween subsets’, in Combinatorics, Computability and Logic (Proceedings of DMTCS’01, Constant¸a, Romania, 2–6 July 2001; C.S. Calude, M.J. Dinneen, S. Sburlan, eds.), 203–214, DMTCS Series 17, Springer Verlag, London, 2001. [26] D.S. Bridges and L.S. Vˆıt¸˘a:‘Characterising near continuity constructively’, Math. Logic Q. 47(4), 535–538, 2001. [27] D.S. Bridges and L.S. Vˆıt¸˘a:‘Cauchy Nets in the Constructive Theory of Apartness Spaces’, Scientiae Math. Jpn. 56, 123–132, 2002. [28] D.S. Bridges and L.S. Vˆıt¸˘a:‘A constructive theory of point-set nearness’, in Topology in Computer Science: Constructivity; Asymmetry and Par- tiality; Digitization (Proc. Dagstuhl Seminar 00231, 4-9 June 2000; R. Kopperman, M. Smyth, D. Spreen, eds.), Theor. Comp. Sci. 305(1–3), 473–489, 2003. [29] D.S. Bridges and L.S. Vˆıt¸˘a: Techniques of Constructive Analysis, Univer- sitext, Springer, New York, 2006. [30] D.S. Bridges and L.S. Vˆıt¸˘a:‘Proximal connectedness’, Fund. Informaticae 83(1-2), 25–34, 2008.

[31] D.S. Bridges and L.S. Vˆıt¸˘a: ‘A constructive theory of apartness on lat- tices’, Scientiae Math. Jpn. 69(2), 187–206, 2009. [32] L.E.J. Brouwer: ‘Over de onbetrouwbaarheid der logische principes’, Ti- jdschrift voor Wijsbegeerte 2, 152–158, 1908. English translation in [34], 107–111. [33] L.E.J. Brouwer: ‘Intuitionistische Zerlegung mathematischer Grundbe- griffe’, Jahresbericht der Deutsche Mathematiker-Vereinigung 33, 241– 256, 1924. [34] L.E.J. Brouwer: Collected Works 1 (A. Heyting, ed.), North-Holland, Am- sterdam, The Netherlands, 1975. [35] P. Cameron, J.G. Hocking, and S.A. Naimpally: Nearness—a better ap- proach to topological continuity and limits, Amer. Math. Monthly 81(7), 739–745, 1974.

[36] R.L. Constable et al.: Implementing Mathematics with the Nuprl Proof Development System, Prentice-Hall, Englewood Cliffs, New Jersey, 1986. [37] L.S. Dediu (Vˆıt¸˘a)and D.S. Bridges: ‘Constructive notes on uniform and locally convex spaces’, in Proceedings of International Symposium FCT ’99 (Ia¸si,Romania), Springer Lecture Notes in Computer Science 1684, 195–203, 1999. 190 References

[38] R. Diaconescu: ‘Axiom of choice and complementation’, Proc. Am. Math. Soc. 51, 176–178, 1975.

[39] H. Diener: ‘Generalising compactness’, Math. Logic Q. 51(1), 49–57, 2008.

[40] M.A.E. Dummett: Elements of Intuitionism (Second Edition), Oxford Logic Guides 39, Clarendon Press, Oxford, 2000.

[41] V.A. Efremoviˇc:‘Infinitesimal spaces’, Dokl. Akad. Nauk. USSR 76, 341– 343, 1951 (in Russian).

[42] R. Engelking: General Topology, Heldermann Verlag, Berlin, Germany, revised and completed edition, 1989.

[43] H.M. Friedman: ‘Set Theoretic Foundations for Constructive Analysis’, Ann. Math. 105(1), 1–28, 1977.

[44] R.I. Goldblatt: Topoi, North-Holland, Amsterdam, The Netherlands, 1979.

[45] N. D. Goodman and J. Myhill: ‘Choice Implies Excluded Middle’, Zeit. Logik und Grundlagen der Math. 24, 461, 1978.

[46] R.J. Grayson: ‘Concepts of general topology in constructive mathematics and in sheaves I’, Ann. Math. Logic 20, 1–41, 1981.

[47] R.J. Grayson: ‘Concepts of general topology in constructive mathematics and in sheaves II’, Ann. Math. Logic 23, 55–98, 1982.

[48] R.S. Havea, H. Ishihara, and L.S. Vˆıt¸˘a:‘Separation properties in neigh- bourhood and quasi-apartness spaces’, Math. Logic Q. 54(1), 58–64, 2008.

[49] S. Hayashi and H. Nakano: PX: A Computational Logic, MIT Press, Cam- bridge, Mass., 1988.

[50] M. Hazewinkel (ed.): Encyclopaedia of Mathematics, Vol. 7, Kluwer Aca- demic, Dordrecht, The Netherlands, 1991.

[51] A. Hedin: ‘A note on set-presentable apartness spaces’, preprint, University of Uppsala, Sweden, 2010.

[52] H. Herrlich: ‘On the extendibility of continuous functions’, Gen. Topol. Appl. 5, 213–215, 1974.

[53] A. Heyting: ‘Die formalen Regeln der intuitionistischen Logik’, Sitzungs- ber. preuss. Akad. Wiss. Berlin, 42–56, 1930.

[54] A. Heyting: Mathematische Grundlagenforschung. Intuitionismus. Beweis- theorie, Springer Verlag, Berlin, 1934. References 191

[55] H. Ishihara: ‘Continuity and nondiscontinuity in constructive mathemat- ics’, J. Symbolic Logic 56(4), 1349–1354, 1991.

[56] H. Ishihara: ‘Markov’s principle, Church’s thesis, and Lindel¨of’s theorem’, Indag. Math. N.S. 4(3), 321–325, 1993.

[57] H. Ishihara: ‘A constructive version of Banach’s inverse mapping theorem’, N.Z. J. Math. 23, 71–75, 1994.

[58] H. Ishihara: ‘Two subcategories of apartness spaces’, Ann, Pure Appl. Logic, to appear.

[59] H. Ishihara, R. Mines, P.M. Schuster, and L.S. Vˆıt¸˘a:‘Quasi-apartness and neighbourhood spaces’, Ann. Pure Appl. Logic 141, 296–306, 2006.

[60] P.T. Johnstone: Stone Spaces, Cambridge Studies in Advanced Mathe- matics 3, Cambridge Univ. Press, Cambridge, U.K., 1982.

[61] P.T. Johnstone: ‘The point of pointless topology’, Bull. Am. Math. Soc. 8(1), 41–53, 1983.

[62] J.L. Kelley: General Topology, van Nostrand, Princeton, New Jersey, 1955; re-published as Graduate Text in Mathematics 27, Springer Verlag, Hei- delberg, Germany, 1975.

[63] A.N. Kolmogorov: ‘Deutung der intuitionistischen Logik’, Math. Z. 35, 58–65, 1932.

[64] B.A. Kushner: Lectures on Constructive Mathematical Analysis, Am. Math. Soc., Providence RI, 1985.

[65] S. Leader: ‘On products of proximity spaces’, Math. Ann. 154, 185–194, 1964.

[66] P. Lietz: From Constructive Mathematics to Computable Analysis via the Realizability Interpretation, Dr. rer. nat. thesis, Technische Universit¨at, Darmstadt, Germany, 2004.

[67] M. Mandelkern: ‘Connectivity of an interval’, Proc. Am. Math. Soc. 54, 170–172, 1976.

[68] P. Martin-L¨of: Notes on Constructive Mathematics, Almqvist and Wiksell, Stockholm, Sweden, 1970.

[69] P. Martin-L¨of:‘Constructive mathematics and computer programming’, in Logic, Methodology, and Philosophy of Science VI (J.J. Cohen, J. Lo´s, H. Pfeiffer, K-P. Podewski, eds), 153–175, North-Holland, Amsterdam, The Netherlands, 1982. 192 References

[70] P. Martin-L¨of: Intuitionistic Type Theory, Notes by G. Sambin, Bibliopolis, Napoli, Italy, 1984.

[71] P. Martin-L¨of:‘An intuitionistic theory of types’, in Twenty-five Years of Constructive Type Theory (G. Sambin, J. Smith, eds), 127–172, Oxford Logic Guides 36, Clarendon Press, Oxford, 1998.

[72] J. Myhill: ‘Constructive set theory’, J. Symbolic Logic 40(3), 1975, 347– 382.

[73] L.J. Nachman: ‘On a conjecture of Leader’, Fund. Math. LXV, 153–155, 1969.

[74] S.A. Naimpally: Proximity Approach to Problems in Topology and Analy- sis, Oldenbourg, Munich, Germany, 2009.

[75] S.A. Naimpally and B.D. Warrack: Proximity Spaces, Cambridge Tracts in Math. and Math. Phys. 59, Cambridge Univ. Press, Cambridge, U.K., 1970.

[76] E. Palmgren and P.M. Schuster: ‘Apartness and formal topology’, N.Z. J. Math. 35, 77–84, 2006.

[77] F. Richman: ‘Constructive mathematics without choice’, in Reuniting the Antipodes—Constructive and Nonstandard Views of the Continuum (P.M. Schuster, U. Berger, and H. Osswald, eds), 199–205, Kluwer, Synthese Library 306, 2001.

[78] F. Riesz: ‘Stetigkeitsbegriff und abstrakte Mengenlehre’, Atti IV Congr. Intern. Mat. Roma II, 18–24, 1908.

[79] G. Sambin: ‘Intuitionistic formal spaces—a first communication’, in Mathematical Logic and its Applications (D.G. Skordev, ed.), 187–204, Plenum, New York, 1987.

[80] G. Sambin: ‘Some points in formal topology’, Theor. Comp. Sci. 305, 347–408, 2003.

[81] G. Sambin: The Basic Picture: Structures for Constructive Topology, Ox- ford Logic Guides, Clarendon Press, Oxford, to appear.

[82] H. Schubert: Topology, Macdonald Technical & Scientific, London, 1968.

[83] P.M. Schuster, L.S. Vˆıt¸˘a,and D.S. Bridges: ‘Apartness as a relation be- tween subsets’, in Combinatorics, Computability and Logic (Proceedings of DMTCS’01, Constant¸a, Romania, 2–6 July 2001; C.S. Calude, M.J. Dinneen, S. Sburlan, eds.), 203–214, DMTCS Series 17, Springer Verlag, London, 2001. References 193

[84] H.A. Schwichtenberg: ‘Program extraction in constructive analysis’, in Logicism, Intuitionism, and Formalism—What has become of them? (S. Lindstr¨om,E. Palmgren, K. Segerberg, and V. Stoltenberg-Hansen, eds), 255–275, Synthese Library 341, Springer Verlag, Berlin, Germany, 2009. [85] L.A. Steen and J.A. Seebach: Counterexamples in Topology, Dover, New York, 1995. [86] T.A. Steinke: Constructive Notions of Compactness in Apartness Spaces, M.Sc. thesis, University of Canterbury, Christchurch, New Zealand, 2011.

[87] A.S. Troelstra: Intuitionistic General Topology, Ph.D. Thesis, University of Amsterdam, The Netherlands, 1966. [88] A.S. Troelstra and D. van Dalen: Constructivism in Mathematics: An In- troduction (two volumes), North-Holland, Amsterdam, The Netherlands, 1988. [89] S.J. Vickers, Topology via Logic, Cambridge Tracts in Theoretical Com- puter Science 5, Cambridge Univ. Press, Cambridge, U.K., 1988. [90] L.S. Vˆıt¸˘aand D.S. Bridges: ‘A constructive theory of point-set nearness’, in Topology in Computer Science: Constructivity; Asymmetry and Par- tiality; Digitization (Proc. Dagstuhl Seminar 00231, 4-9 June 2000; R. Kopperman, M. Smyth, D. Spreen, eds.), 473–489, Theor. Comp. Sci. 305(1–3), 2003. [91] F.A. Waaldijk: Constructive Topology, Ph.D. thesis, University of Ni- jmegen, The Netherlands, 1996.

[92] K. Weihrauch: Computable Analysis, EATCS Texts in Theoretical Com- puter Science, Springer Verlag, Heidelberg, Germany, 2000. [93] S. Willard: General Topology, Addison-Wesley, Reading, Mass., 1970. INDEX

A4s, 68 BHK interpretation, 3 A4ss, 68 binary relation, 11 absolute value, 9 BISH, 2 apart, 20, 66 Bishop’s lemma, 134 apartness class of uniformities, 157 Bishop, Errett, vii, 17 apartness space, 67 bounded mapping, 10 apartness space, (quasi-)uniform, 86 Brouwer, L.E.J., 17 apartness space, product (set-set), Brouwerian counterexample, 6 145 apartness subspace, 70 Cartesian product, 10 apartness topology Cauchy net, neatly, 173 topology, apartness, 31 Cauchy net, uniformly, 123 apartness, induced, 23 Cauchy sequence, 14 apartness, induced by quasi-uniformity, Cauchy sequence, CHN-, 120, 178 87 Cauchy sequence, metrically, 121 apartness, on frames, 185 choice, axiom of, 12 apartness, point-point, 19 choice, countable, 12 apartness, point-set, 21 choice, dependent, 12 apartness, product (set-set), 144 Church–Markov–Turing thesis, 18 apartness, quasi-metric, 22 CLASS, 18 apartness, set-set, 67 classical mathematics, 18 apartness, topological, 28 closed set, 14, 16 approximation, ε-, 14 closure, 14, 16 at most n elements, 11 cofinal, 125 cofinite, 47 B7, 86 compact, 181 ball, closed, 13 compact metric space, 14 ball, open, 13 compact, (uniformly), 130 base for a topology, 16 compact, filter, 18 base of entourages, 84 compact, neatly, 171 basic neighbourhood, 184 compact, open-cover, 18

194 Index 195 compact, sequentially, 18 EF, 69 complement, 20 Efremoviˇcproperty, 69 complement, apartness, 20 empty, 9 complement, logical, 19 entourage, 78 complete metric space, 14 entourage, symmetric, 78 complete, neatly, 173 equal sets, 9 complete, totally sequentially, 135 equality, 8 complete, uniformly, 128 essentially nonconstructive, 4, 5, 7 completely regular, 39 eventually, 98 composite, 10 eventually bounded away from, 44 composition, 10 eventually close, 98 computable analysis, Weihrauch’s, exactly n elements, 12 18 extension, 10 connectives, 2 extensional, 8 constructive mathematics, vii, 17 continuous, 36 factor, 10 continuous, nearly, 36 family, 11 continuous, strongly, 92 family, intersection of, 11 continuous, topologically, 16, 36 family, union of, 11 continuous, uniform sequentially, 98 filter, 77 continuous, uniformly, 14, 92 filter base, 77 convergent net, 41 finite, 11 convergent sequence, 14 finitely enumerable, 11 convergent, apartness, 41 first countable, 134 convergent, filter, 130 formal topology, x, 184 convergent, neatly, 173 frame, x, 183 convergent, pointwise, 107 function, 9 convergent, proximally, 107 function, point, 184 convergent, topologically, 42 convergent, uniformly, 108 Goldbach Conjecture, 3 convergent, uniformly sequentially, greatest lower bound, 6 108 cotransitive, 47 Hausdorff (apartness), 43 cotransitivity, 4 Hausdorff (topologically), 43 countable, 12 Heyting, Arend, 17 covering relation, 184 impredicativity, 61 dense, 16 index set, 11, 40 dense, strongly, 133 inequality relation, 8 diagonal, 78 inequality, compatible, 15 Diener, Hannes, 181 inequality, denial, 9 directed set, 40 inequality, induced, 23 discrete set, 9 inequality, nontrivial, 8 disjoint, 11 inequality, standard, 9 domain, 9 inequality, tight, 8 196 Index inequality, triangle, 13 mapping, 9 infimum, 6, 10 mapping, approximable, 184 infinite, 12 Markov’s principle, 5 infinitely often, 98 Markov’s principle, weak disjunctive, inhabited, 6 59 injective, 10 Martin-L¨of,Per, 184 INT, 17 metric, 13 interior, 14, 16 metric, Euclidean, 13 intuitionistic logic, axioms, 17 Morse set theory, 62 intuitionistic mathematics, 17 MP, 5 Ishihara, Hajime, 63 MPor, 59

Kolmogorov, Andrey, 17 n-chain, 80 Kripke model, 18, 64 near, 24 nearly closed, 35 law of excluded middle, 2 nearly open, 31 least upper bound, 6 neat cover, 166 least-upper-bound principle, classi- neighbourhood, 16 cal, 6 neighbourhood base, 16 least-upper-bound principle, construc- neighbourhood space, 183 tive, 7 nested neighbourhoods property, 127 left uniform structure, 90 net, 40 LEM, 2 net, basic neighbourhood, 41 length of a sequence, 11 net, basic punctured neighbourhood, lesser limited principle of omniscience, 41 5 nonempty, 6 limit, 14, 41 nontrivial seminormed space, 15 limit of a function, 49 nonzero vector, 15 limited principle of omniscience, 4 norm, 16 LLPO, 5 normed space, 16 locale, x, 184 locally compact, 15 one-one, 10 locally convex space, 29 onto, 10 locally decomposable, 34 open set, 14, 16 locally decomposable, topologically, 33 partial order, 64 locally totally bounded, 15 power set, 61 located, 15 pre-apartness space, T1, 26 located, almost, 136 pre-apartness subspace, 70 located, neatly, 167 pre-apartness, (quasi-)uniform, 81 located, weakly, 134 pre-apartness, induced, 70 Lodato, 21 pre-apartness, left, 75 LPO, 4 pre-apartness, point-set, 20 pre-apartness, right, 177 map, 9 pre-apartness, set-set, 66 Index 197 pre-apartness, symmetric, 66 sequence, finite, 11 pre-apartness, topological, 28 set, 7 pre-uniform structure, 113 stable, 151 preorder, 40 strong extensionality, 10 preorder, reverse inclusion, 41 strong unique limits property, 44 product (quasi-)uniform structure, strongly generates, 163 94 subbase, 16 product of topological spaces, 16 subfinite, 18 product pre-apartness, 52 subnet, 125 product pre-apartness space, 52 subsequence, 11 product quasi-metric, 13 subset, 8 product topology, 16 subspace, 16 projection, 11 subspace, (pre-)apartness, 23 proximally connected, 152 subspace, metric, 13 proximity, 21 subspace, uniform, 78 pseudometric, 178 sufficiently large, 98 SULP, 44 quantifiers, 2 superset, 8 quasi-metric, 13 supremum, 6, 10 quasi-uniform space, 78 quasi-uniform structure, 78 T pre-apartness space, 26 quasi-uniform structure, product, 94 1 T topological space, 91 quasi-uniform structure, subspace, 1 term, 11 78 topological A5, 28 quasi-uniformity, 78 quasi-uniformity, compatible, 87 topological group, 74 topological group, decomposable, 76 range, 10 topological space, 16 real number, 4 topologically consistent, 32 recursive constructive mathematics, topology, 16 18 topology, coarser, 16 relation between elements, 11 topology, cofinite, 47 restriction, 10 topology, denial, 60 reverse Kolmogorov property, 23, 63 topology, finer, 16 reverse Kolmogorov property, topo- topology, locally convex, 29 logical, 29 topology, of proximal convergence, Richman, Fred, 12 111 RUSS, 18 topology, of uniform convergence, 114 Sambin, Giovanni, 184 topology, point-free, 183 second-countable, 33 topology, quasi-uniform, 80 seminorm, 15 topology, subspace, 16 separable, 16 totally bounded, 14 separation, first axiom of, 29 totally bounded uniform space, 95 sequence, 11 totally bounded, locally, 179 198 Index totally bounded, weak-locally, 143, 179 totally Cauchy net, 121 totally complete, 128 trichotomy, 4 trichotomy, weak form, 5 tuple, 11 type, 7, 64

U-approximation, 95 U-small set, 95 ULP, 44 ultrafilter, 130 uniform equality, 78 uniform space, 78 uniform structure, 78 uniform structure, compatible, 162 uniform structure, generated, 162 uniformity, 78 unique limits property, 44 upper order located, 7 weak limited principle of omniscience, 2 weak nested neighbourhoods prop- erty, 39 weakly symmetrically separated, 121 well constructed, 62 well contained, 157 WLPO, 2 WNN, 39

Y X , 10