Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

The point of pointless topology


Author: Peter T. Johnstone
Journal: Bull. Amer. Math. Soc. 8 (1983), 41-53
MSC (1980): Primary 06A23, 18B30, 54A05; Secondary 01A60, 06D05, 18B25
DOI: https://doi.org/10.1090/S0273-0979-1983-15080-2
MathSciNet review: 682820
Full-text PDF

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • 1. B. Banaschewski, The duality of distributive continuous lattices, Canad. J. Math. 32 (1980), no. 2, 385–394. MR 571932, https://doi.org/10.4153/CJM-1980-030-3
  • 2. B. Banaschewski, Extension of invariant linear functionals: Hahn-Banach in the topos of 𝑀-sets, J. Pure Appl. Algebra 17 (1980), no. 3, 227–248. MR 579085, https://doi.org/10.1016/0022-4049(80)90047-X
  • 3. B. Banaschewski and C. J. Mulvey, Stone-Čech compactification of locales. I, Houston J. Math. 6 (1980), no. 3, 301–312. MR 597771
  • 4. J. Bénabou, Treillis locaux et paratopologies, Séminaire Ehresmann (Topologie et Géométrie Différentielle), 1re année (1957-58), exposé 2. MR 121774
  • 5. C. H. Dowker and D. Papert, Quotient frames and subspaces, Proc. London Math. Soc. 16 (1966), 275-296. MR 202648
  • 6. C. H. Dowker and D. Papert, On Urysohn's lemma, General Topology and its Relations to Modern Analysis and Algebra (Proc. Prague Symposium, 1966), Academia, Prague, 1967, pp. 111-114. MR 238744
  • 7. C. H. Dowker and D. Strauss, Separation axioms for frames, Colloq. Math. Soc. János Bolyai 8 (1972), 223-240. MR 394559
  • 8. C. H. Dowker and D. Strauss, Paracompact frames and closed maps, Sympos. Math. 16 (1975), 93-116. MR 410663
  • 9. C. H. Dowker and D. Strauss, Sums in the category of frames, Houston J. Math. 3 (1977), 7-15. MR 442900
  • 10. R. Dyckhoff, Factorization theorems and projective spaces in topology, Math. Z. 127 (1972), 256-264. MR 312452
  • 11. R. Dyckhoff, Categorical methods in dimension theory, Categorical Topology (Proc. Mannheim Conf., 1975), Lecture Notes in Math., vol. 540, Springer-Verlag, Berlin and New York, 1976, pp. 220-242. MR 448318
  • 12. C. Ehresmann, Gattungen von lokalen Strukturen, Jber. Deutsch. Math.-Verein. 60 (1957), 59-77. MR 95894
  • 13. C. Ehresmann, Catégories topologiques et catégories diffèrentiables, Colloq. Géom. Diff. Globale (Bruxelles), Centre Belge Rech. Math., Louvain, 1959, pp. 137-150. MR 116360
  • 14. J. Fingerman, The historical and philosophical significance of the emergence of point-set topology, Ph.D. thesis, Univ. of Chicago, 1981.
  • 15. M. P. Fourman and J. M. E. Hyland, Sheaf models for analysis, Applications of sheaves (Proc. Res. Sympos. Appl. Sheaf Theory to Logic, Algebra and Anal., Univ. Durham, Durham, 1977) Lecture Notes in Math., vol. 753, Springer, Berlin, 1979, pp. 280–301. MR 555550
  • 16. R. H. Fox, Covering spaces with singularities, Algebraic Topology (Sympos. in honor of S. Lefschetz), Princeton Univ. Press, Princeton, N. J., 1957, pp. 243-257. MR 123298
  • 17. M. Fréchet, Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo 22 (1906), 1-71.
  • 18. A. M. Gleason, Projective topological spaces, Illinois J. Math. 2 (1958), 482-489. MR 121775
  • 19. R. J. Grayson, Concepts of general topology in constructive mathematics and in sheaves. II, Ann. Math. Logic 23 (1982), no. 1, 55–98. MR 674673, https://doi.org/10.1016/0003-4843(82)90010-9
  • 20. J. D. Halpern, The independence of the axiom of choice from the Boolean prime ideal theorem, Fund. Math. 55 (1964), 57-66. MR 164891
  • 21. F. Hausdorff, Grundzüge der Mengenlehre, Veit & Co., Leipzig, 1914. MR 31025
  • 22. M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43-60. MR 251026
  • 23. Karl H. Hofmann and Jimmie D. Lawson, The spectral theory of distributive continuous lattices, Trans. Amer. Math. Soc. 246 (1978), 285–310. MR 515540, https://doi.org/10.1090/S0002-9947-1978-0515540-7
  • 24. J. M. E. Hyland, Function-spaces in the category of locales, Continuous Lattices (Proc. Bremen workshop, 1979), Lecture Notes in Math., vol. 871, Springer-Verlag, Berlin and New York, 1981, pp. 264-281.
  • 25. J. R. Isbell, Atomless parts of spaces, Math. Scand. 31 (1972), 5-32. MR 358725
  • 26. John Isbell, Product spaces in locales, Proc. Amer. Math. Soc. 81 (1981), no. 1, 116–118. MR 589150, https://doi.org/10.1090/S0002-9939-1981-0589150-5
  • 27. I. M. James, Ex-homotopy theory. I, Illinois J. Math. 15 (1971), 324-337. MR 296945
  • 28. I. M. James, Alternative homotopy theories, Enseign. Math. (2) 23 (1977), 221-237. MR 482752
  • 29. P. T. Johnstone, Topos theory, L.M.S. Math. Mono., no. 10, Academic Press, London, 1977. MR 470019
  • 30. Peter T. Johnstone, Open maps of toposes, Manuscripta Math. 31 (1980), no. 1-3, 217–247. MR 576498, https://doi.org/10.1007/BF01303275
  • 31. P. T. Johnstone, The Gleason cover of a topos. I, J. Pure Appl. Algebra 19 (1980), 171–192. MR 593253, https://doi.org/10.1016/0022-4049(80)90100-0
  • 32. P. T. Johnstone, The Gleason cover of a topos. II, J. Pure Appl. Algebra 22 (1981), no. 3, 229–247. MR 629332, https://doi.org/10.1016/0022-4049(81)90100-6
  • 33. P. T. Johnstone, Tychonoff’s theorem without the axiom of choice, Fund. Math. 113 (1981), no. 1, 21–35. MR 641111, https://doi.org/10.4064/fm-113-1-21-35
  • 34. Peter T. Johnstone, Factorization theorems for geometric morphisms. II, Categorical aspects of topology and analysis (Ottawa, Ont., 1980) Lecture Notes in Math., vol. 915, Springer, Berlin-New York, 1982, pp. 216–233. MR 659894
  • 35. Peter T. Johnstone, Stone spaces, Cambridge Studies in Advanced Mathematics, vol. 3, Cambridge University Press, Cambridge, 1982. MR 698074
  • 36. P. T. Johnstone, The Vietoris monad on the category of locales (in preparation).
  • 37. André Joyal and Myles Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 51 (1984), no. 309, vii+71. MR 756176, https://doi.org/10.1090/memo/0309
  • 38. F. C. Kirwan, Uniform locales, Part III dissertation, Univ. of Cambridge, 1981.
  • 39. K. Kuratowski, Sur l'opération Ā de l'analysis situs, Fund. Math. 3 (1922), 182-199.
  • 40. W. Kuyk, Complementarity in mathematics, Math, and its applications, vol. 1, Reidel, Dordrecht, 1977. MR 465765
  • 41. F. W. Lawvere, Continuously variable sets: Algebraic geometry = geometric logic (Logic Colloq., 1973), Studies in Logic and the Foundations of Math., vol. 80, North-Holland, Amsterdam, 1975, pp. 135-156. MR 392567
  • 42. J. Leray, L'anneau spectral et l'anneau filtré d'homologie d'un espace localement compact et d'une application continue, J. Math. Pures Appl. 29 (1950), 1-80. MR 37505
  • 43. F. E. J. Linton, Some aspects of equational categories, Categorical Algebra (Proc. La Jolla Conf., 1965), Springer-Verlag, Berlin and New York, 1966, pp. 84-94. MR 209335
  • 44. J. Łoš and C. Ryll-Nardzewski, On the application of Tychonoff's theorem in mathematical proofs, Fund. Math. 38 (1951), 233-237. MR 48795
  • 45. S. Mac Lane, The influence of M. H. Stone on the origins of category theory, Functional Analysis and Related Fields, Springer-Verlag, Berlin and New York, 1970, pp. 228-241. MR 263888
  • 46. S. Mac Lane, Sets, topoi and internal logic in categories (Logic Colloq., 1973), Studies in Logic and the Foundations of Math., vol. 80, North-Holland, Amsterdam, 1975, pp. 119-134. MR 384545
  • 47. J. C. C. McKinsey and A. Tarski, The algebra of topology, Ann. of Math. (2) 45 (1944), 141-191. MR 9842
  • 48. J. C. C. McKinsey and A. Tarski, On closed elements in closure algebras, Ann. of Math. (2) 47 ( 1946), 122-162. MR 15037
  • 49. J. Nagata, Modern dimension theory, Noordhoff, Groningen, 1965.
  • 50. S. B. Niefield, Exactness and projectivity, Category theory (Gummersbach, 1981) Lecture Notes in Math., vol. 962, Springer, Berlin-New York, 1982, pp. 221–227. MR 682960
  • 51. G. Nöbeling, Grundlagen der analytischen Topologie, Die Grundlehren der math. Wissenschaften, Band 72, Springer-Verlag, Berlin and New York, 1954. MR 68197
  • 52. D. Papert, Lattices of functions, measures and point-sets, Ph.D. thesis, Univ. of Cambridge, 1958.
  • 53. S. Papert, The lattices of logic and topology, Ph.D. thesis, Univ. of Cambridge, 1959.
  • 54. H. Simmons, A couple of triples, Topology Appl. 13 (1982), no. 2, 201–223. MR 644114, https://doi.org/10.1016/0166-8641(82)90021-9
  • 55. M. H. Stone, Boolean algebras and their applications to topology, Proc. Nat. Acad. Sci. U.S.A. 20 (1934), 197-202.
  • 56. M. H. Stone, The theory of representations for Boolean algebras, Trans. Amer. Math. Soc. 40 (1936), no. 1, 37–111. MR 1501865, https://doi.org/10.1090/S0002-9947-1936-1501865-8
  • 57. M. H. Stone, Topological representation of distributive lattices and Brouwerian logics, Časopis pešt. mat. fys. 67 (1937), 1-25.
  • 58. Leopold Vietoris, Bereiche zweiter Ordnung, Monatsh. Math. Phys. 32 (1922), no. 1, 258–280 (German). MR 1549179, https://doi.org/10.1007/BF01696886
  • 59. Henry Wallman, Lattices and topological spaces, Ann. of Math. (2) 39 (1938), no. 1, 112–126. MR 1503392, https://doi.org/10.2307/1968717
  • 60. S. Willard, General topology, Addison-Wesley, Reading, Mass., 1970. MR 264581
  • 61. Gavin C. Wraith, Galois theory in a topos, J. Pure Appl. Algebra 19 (1980), 401–410. MR 593261, https://doi.org/10.1016/0022-4049(80)90108-5
  • 62. Gavin C. Wraith, Localic groups, Cahiers Topologie Géom. Différentielle 22 (1981), no. 1, 61–66. Third Colloquium on Categories (Amiens, 1980), Part II. MR 609160
  • 63. A. Zarelua, Sheaf theory and zero-dimensional mappings, Applications of sheaves (Proc. Res. Sympos. Appl. Sheaf Theory to Logic, Algebra and Anal., Univ. Durham, Durham, 1977) Lecture Notes in Math., vol. 753, Springer, Berlin, 1979, pp. 768–779. MR 555572

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1980): 06A23, 18B30, 54A05, 01A60, 06D05, 18B25

Retrieve articles in all journals with MSC (1980): 06A23, 18B30, 54A05, 01A60, 06D05, 18B25


Additional Information

DOI: https://doi.org/10.1090/S0273-0979-1983-15080-2

American Mathematical Society