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Spaces of Mappings and Rings of Continuous Functions

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General Topology III

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 51))

Abstract

Passing to the limit in a set of functions is one of the most frequently encountered phenomena in mathematics. We meet it in the theory of differential equations, functional analysis, algebraic topology, differential geometry, probability theory, and in the various applications of these and other areas of mathematics.

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References

  • Alexandroff, P.S., Pasynkov, B.A. (1973): Introduction to Dimension Theory. Nauka, Moscow. Zbl. 272. 54028

    Google Scholar 

  • Amir, D., Lindenstraiiss, J. (1968): The structure of weakly compact sets in Banach spaces. Ann. Math., II. Ser. 88, No. 1, 35–46. Zbl. 164, 149

    Google Scholar 

  • Arens, R., Dugundji, J. (1951): Topologies for function spaces. Pac. J. Math. 1, No. 1, 5–31. Zbl. 44, 118

    Google Scholar 

  • Arhangel’skii, A.V. (= Arkhangel’skii, A.V.)(1959): An addition theorem for the weight of sets lying in bicompacta. Dokl. Akad. Nauk SSSR 126, No. 2, 239–241 (in Russian). Zbl. 87, 376

    Google Scholar 

  • Arhangel’skii, A.V. (1976): On certain topological spaces encountered in functional analysis. Usp. Mat. Nauk 31, No. 5, 17–32.

    MathSciNet  Google Scholar 

  • Arhangel’skii, A.V. (1976): English transi.: Russ. Math. Surv. 31, No. 5, 14–30. Zbl. 344. 46058

    Google Scholar 

  • Arhangel’skii, A.V. (1982): On linear homeomorphisms of function spaces. Dokl. Akad. Nauk SSSR 264, No. 6, 1289–1292.

    MathSciNet  Google Scholar 

  • Arhangel’skii, A.V. (1982): English transi.: Soviet Math., Dokl. 25, 852–855. Zbl. 522. 54015

    Google Scholar 

  • Arhangel’skii, A.V. (1984a): Continuous maps, factorization theorems and function spaces. Tr. Mosk. Mat. 0: va 47, 3–21.

    Google Scholar 

  • Arhangel’skii, A.V. (1984a): English transi.: Trans. Mosc. Math. Soc. 1985, 1–22 (1985). Zbl. 588. 54018

    Google Scholar 

  • Arhangel’skii, A.V. (1984b): Function spaces in the topology of pointwise convergence and compacta. Usp. Mat. Nauk 39, No. 5, 11–50.

    Google Scholar 

  • Arhangel’skii, A.V. (1984b): English transl.: Russ. Math. Surv. 39, No. 5, 9–56. Zbl. 568. 54016

    Google Scholar 

  • Arhangel’skii, A.V. (1985): Function spaces in the topology of pointwise convergence. Part I, in: General Topology: Function Spaces and Dimension, Collect. Work, Moscow 3–66 (in Russian). Zbl. 638. 54015

    Google Scholar 

  • Arhangel’skii, A.V. (1987): A survey of Cr-theory. Quest. Answers General Topology 5, 1–109 (in Russian). Zbl. 634. 54012

    Google Scholar 

  • Arhangel’skii, A.V. (1988): Some results and problems in Cr-theory. Proc. 6th Symp., Prague/Czech 1986, Reg. Expo. Math. 16, 11–31. Zbl. 658. 54013

    Google Scholar 

  • Arhangel’skii, A.V. (1989a): Topological Function Spaces. Izdat. Mosk. Gos. Univ., Moscow (in Russian) Zbl. 781. 54014

    Google Scholar 

  • Arhangel’skii, A.V. (1989b): General Topology II: Compactness. Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. Fundam. Napravleniya 50, 5–128.

    Google Scholar 

  • Arhangel’skii, A.V. (1989b): English transl. in: Encycl. Math. Sci. 50. Springer-Verlag, Berlin Heidelberg New York 1995. Zbl. 709. 54018

    Google Scholar 

  • Arhangel’skii, A., Choban, M.M. (1988): The extension property of Tychonoff spaces and generalized retracts. C. R. Acad. Bulg. Sci. 41, No. 12, 5–7. Zbl. 675. 54014

    Google Scholar 

  • Arhangel’skii, A.V., Ponomarev, V.I. (1974): Fundamentals of General Topology in Problems and Exercises. Nauka, Moscow (in Russian). Zbl. 287. 54001

    Google Scholar 

  • Arhangel’skii, A.V., Tkachuk, V.V. (1985): Function Spaces and Topological Invariants. Izdat Mosk. Gos. Univ., Moscow (in Russian)

    Google Scholar 

  • Baars, J., de Groot, J. (1988): An isomorphical classification of function spaces of zero-dimensional locally compact separable metric spaces. Commentat. Math. Univ. Carol. 29, No. 3, 577–595. Zbl. 684. 54011

    Google Scholar 

  • Banach, S. (1932): Théorie des Opérations Linéaires. PWN, Warsaw. Zbl. 5,209 Baturov, D.P. (1987): Subspaces of function spaces. Vestn. Mosk. Univ., Ser. I, No. 4, 66–69.

    Google Scholar 

  • Banach, S. (1932): English transl.: Mosc. Univ. Math. Bull. 42, No.4, 75–78 (1987). Zbl. 626. 54022

    Google Scholar 

  • Benyamini, Y., Rudin, M.E., Wage, M. (1977): Continuous images of weakly compact subsets of Banach spaces. Pac. J. Math. 70, No. 2, 309–324. Zbl. 374. 46011

    Google Scholar 

  • Borges, C.J.R. (1966): On stratifiable spaces. Pac. J. Math. 17, No. 1, 1–16. Zbl. 175, 198

    Google Scholar 

  • Choban, M.M., Dodon, N.K. (1979): Theory of P-Scattered Spaces. Shtiintsa, Kishinev (in Russian). Zbl. 506. 54029

    Google Scholar 

  • Christensen, J.P.R. (1981): Joint continuity of separately continuous functions. Proc. Am. Math. Soc. 82, No. 3, 455–461. Zbl. 472. 54007

    Google Scholar 

  • Corson, H.H. (1959): Normality in subsets of product spaces. Am. J. Math. 81, 785–796. Zbl. 95, 373

    Google Scholar 

  • Dijkstra, J., Grillot, T., Lutzer, D., van Mill, J. (1985): Function spaces of low Borel complexity. Proc. Am. Math. Soc. 94, No. 4, 703–710. Zbl. 564. 54007

    Google Scholar 

  • van Douwen, E.K. (1975): Simultaneous linear extension of continuous functions. General Topology Appl. 5, 297–319. Zbl. 309. 54013

    Google Scholar 

  • Dranishnikov, A.N. (1986): Absolute F-valued retracts and function spaces in the topology of pointwise convergence. Sib. Mat. Zh. 27, No. 3, 74–86.

    MathSciNet  Google Scholar 

  • Dranishnikov, A.N. (1986): English transl.: Sib. Math. J. 27, 366–376 (1976). Zbl. 622. 54027

    Google Scholar 

  • Dugundji, J. (1966): Topology. Allyn and Bacon, Inc. XVI. Zbl. 144, 215

    Google Scholar 

  • Engelking, R. (1977): General Topology. PWN, Warsaw. Zbl. 373. 54001

    Google Scholar 

  • Filippov, V.V. (1980): On the existence of solutions of differential equations. Colloq. Math. Soc. Janos Bolyai 23. North-Holland, New York, 397–411. Zbl. 433. 34005

    Google Scholar 

  • Fréchet, M. (1906): Sur quelques points du calcul fonctionnel. Rend. Circ. Mat. Palermo 22, 1–74. FdM. 37, 348

    Google Scholar 

  • Gel’fand, I.M., Kolmogorov, A.N. (1939): On rings of continuous functions on topological spaces. Dokl. Akad. Nauk SSSR 22, No. 1, 11–15. Zbl. 21, 411

    Google Scholar 

  • Gillman, L., Jerison, M. (1960): Rings of Continuous Functions. van Nostrand, New York. Zbl. 93, 300

    Google Scholar 

  • Gul’ko, S.P. (1979): On the structure of spaces of continuous functions and their hereditary paracompactness. Usp. Mat. Nauk 34, No. 6, 33–40.

    MathSciNet  MATH  Google Scholar 

  • Gul’ko, S.P. (1979): English transi.: Russ. Math. Surv. 34, No. 6, 36–44. Zbl. 421. 46020

    Google Scholar 

  • Gul’ko, S.P., Khmylëva, T.E. (1986): Compactness is not preserved under the t-equivalence relation. Mat. Zametki 39, No. 6, 895–903.

    MathSciNet  Google Scholar 

  • Gul’ko, S.P., Khmylëva, T.E. (1986): English transi.: Math. Notes 39, 484–488. Zbl. 629. 54007

    Google Scholar 

  • Heath, R.W., Lutzer, D.J., Zenor, P.L. (1973): Monotonically normal spaces. Trans. Am. Math. Soc. 178, 481–493. Zbl. 269. 54009

    Google Scholar 

  • Hewitt, E. (1948): Rings of real-valued continuous functions. I. Trans. Am. Math. Soc. 64, 45–99. Zbl. 32, 286

    Google Scholar 

  • Hochster, M. (1969): Prime ideal structure in commutative rings. Trans. Am. Math. Soc. 142, 43–60. Zbl. 184, 294

    Google Scholar 

  • Isbell, J.R. (1964): Uniform Spaces. Math. Surv. 12. Providence, RI. Am. Math. Soc. XI. Zbl. 124, 156

    Google Scholar 

  • Kadets, M.I. (1967): Proof of the topological equivalence of all separable infinite-dimensional Banach spaces. Funkts. Anal. Prilozh. 1, No. 1, 61–70

    MATH  Google Scholar 

  • Kadets, M.I. (1967): English transi: Funct. Anal Appl. 1, No. 1, 53–62. Zbl. 166, 106

    Google Scholar 

  • Kelley, J.L. (1955): General Topology. van Nostrand, New York. Zbl. 66, 166

    Google Scholar 

  • Khelemskij, A.Ya. (1986): Homology in Banach and Topological Algebras. Izdat. Mosk. Gos. Univ., Moscow. Zbl. 608. 46046

    Google Scholar 

  • Kolmogorov, A.N., Fomin, S.V. (1968): Elements of the Theory of Functions and Functional Analysis. Nauka, Moscow (in Russian). Zbl. 167, 421

    Google Scholar 

  • Lutzer, D., van Mill, J., Pol, R. (1985): Descriptive complexity of function spaces. Trans. Am. Math. Soc. 291, No. 1, 121–128. Zbl. 574. 54042

    Google Scholar 

  • McCoy, R.A., Ntantu, I. (1988): Topological properties of spaces of continuous functions. Lect. Notes Math. 1315. Zbl. 674. 54001

    Google Scholar 

  • Michael, E. (1966): lao-spaces. J. Math. Mech. 15, 983–1002. Zbl. 148, 167

    Google Scholar 

  • van Mill, J. (1982): A homogeneous Eberlein compact space which is not metrizable. Pac. J. Math. 101, No. 1, 141–146. Zbl. 495. 54020

    Google Scholar 

  • Nagata, J. (1949): On lattices of functions on topological spaces and of functions on uniform spaces. Osaka J. Math. 1, No. 2, 166–181. Zbl. 36, 386

    Google Scholar 

  • Namioka, I. (1974): Separate continuity and joint continuity. Pac. J. Math. 51, 515–531. Zbl. 294. 54010

    Google Scholar 

  • Negrepontis, S. (1984): Banach spaces and topology, in: Handbook of Set-theoretic Topology. North-Holland, New York, 1045–1142. Zbl. 584. 46007

    Google Scholar 

  • Oxtoby, J.C. (1971): Measure and Category. Springer-Verlag, New York. Zbl. 217, 92

    Google Scholar 

  • Pavlovskij, D.S. (1980): Spaces of continuous functions. Dokl. Akad. Nauk SSSR 253, No. 1, 38–41.

    MathSciNet  Google Scholar 

  • Pavlovskij, D.S. (1980): English transl.: Sov. Math., Dokl. 22, 34–37. Zbl. 489. 54011

    Google Scholar 

  • Pelczynski, A. (1968): Linear Extensions, Linear Averagings, and their Applications. Diss. Math. 58. Zbl. 165, 146

    Google Scholar 

  • Pestov, V.G. (1982): Coincidence of the dimensions dim of /-equivalent spaces. Dokl. Akad. Nauk SSSR 266, No. 3, 553–556.

    MathSciNet  Google Scholar 

  • Pestov, V.G. (1982): English transl.: Sov. Math., Dokl. 26, 380–383] Zbl. 518. 54030

    Google Scholar 

  • Pol, R. (1986): Note on compact sets of first Baire class functions. Proc. Am. Math. Soc. 96, No. 1, 152–154. Zbl. 584. 54014

    Google Scholar 

  • Preiss, D., Simon, P. (1974): A weakly pseudocompact subspace of Banach space is weakly compact. Commentat. Math. Univ. Carol. 15, No. 4, 603–609. Zbl. 306. 54033

    Google Scholar 

  • Roitman, J. (1984): Basic S and L, in: Handbook of Set-theoretic Topology. North-Holland, New York, 295–326. Zbl. 594. 54001

    Google Scholar 

  • Rosenthal, H.P. (1970): On injective Banach spaces and the spaces L°°(µ) for finite measures A. Acta Math. 124, No. 3, 205–248. Zbl. 207, 428

    Google Scholar 

  • Rosenthal, H.P. (1977): Pointwise compact subsets of the first Baire class. Am. J. Math. 99, No. 2, 362–378. Zbl. 392. 54009

    Google Scholar 

  • Rudin, W. (1973): Functional Analysis. McGraw-Hill, New York. Zbl. 253. 46001

    Google Scholar 

  • Semadeni, Z. (1971): Banach Spaces of Continuous Functions. I. PWN, Warsaw. Zbl. 225. 46030

    Google Scholar 

  • Shchepin, E.V. (1977): A finite-dimensional bicompact absolute neighbourhood retract is metrizable. Dokl. Akad. Nauk SSSR 233, No. 2, 304–307.

    MathSciNet  Google Scholar 

  • Shchepin, E.V. (1977): English transl.: Sov. Math., Dokl. 18, 402–406. Zbl. 372. 54031

    Google Scholar 

  • Sikorski, R. (1964): Boolean Algebras. Springer-Verlag, New York. Zbl. 123, 13

    Google Scholar 

  • Sokolov, G.A. (1986): Lindelöf spaces of continuous functions. Mat. Zametki 39, No. 6, 887–894.

    MathSciNet  Google Scholar 

  • Sokolov, G.A. (1986): English transl: Math. Notes 39, 480–484. Zbl. 607. 54012

    Google Scholar 

  • Stone, M.H. (1937): Applications of the theory of Boolean rings to general topology. Trans. Am. Math. Soc. 41, 375–481. Zbl. 17, 135

    Google Scholar 

  • Stone, M.H. (1947): The generalized Weierstrass approximation theorem. Math. Mag. 21, 167–184, 237–254

    Google Scholar 

  • Talagrand, M. (1979): Espaces de Banach faiblement K-analytiques. Ann. Math., II. Ser. 110, No. 3, 407–438. Zbl. 414. 46015

    Google Scholar 

  • Tall, F.D. (1984): Normality versus collectionwise normality, in: Handbook of Set-theoretic Topology. North-Holland, New York, 685–732. Zbl. 552. 54011

    Google Scholar 

  • Tkachuk, V.V. (1983): Cardinal invariants of Souslin number type. Dokl. Akad. Nauk SSSR 270, No. 4, 795–798.

    MathSciNet  Google Scholar 

  • Tkachuk, V.V. (1983): English transi.: Soy. Math., Dokl. 27, 681–684. Zbl. 539. 54001

    Google Scholar 

  • Torunczyk, H. (1981): Characterizing Hilbert space topology. Fundam. Math. 111, No. 3, 247–262. Zbl. 468. 57015

    Google Scholar 

  • Uspenskij, V.V. (1983): On the topology of a free locally convex space. Dokl. Akad. Nauk SSSR 270, No. 6, 1334–1337.

    MathSciNet  Google Scholar 

  • Uspenskij, V.V. (1983): English transl.: Sov. Math., Dokl. 27, 781785. Zbl. 569. 54015

    Google Scholar 

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Authors
  1. A. V. Arhangel’skii
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Editors and Affiliations

  1. Department of Mathematics, University of Moscow, Leninskie Gory, 119899, Moscow, Russia

    A. V. Arhangel’skii

  2. Department of Mathematics, Ohio University, 3221 Morton Hall, 45701-2979, Athens, Ohio, USA

    A. V. Arhangel’skii

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Arhangel’skii, A.V. (1995). Spaces of Mappings and Rings of Continuous Functions. In: Arhangel’skii, A.V. (eds) General Topology III. Encyclopaedia of Mathematical Sciences, vol 51. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-07413-8_2

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  • DOI: https://doi.org/10.1007/978-3-662-07413-8_2

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