Completely Regular Bishop Spaces

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9136)

Abstract

Bishop’s notion of a function space, here called a Bishop space, is a constructive function-theoretic analogue to the classical set-theoretic notion of a topological space. Here we introduce the quotient, the pointwise exponential and the completely regular Bishop spaces. For the latter we present results which show their correspondence to the completely regular topological spaces, including a generalized version of the Tychonoff embedding theorem for Bishop spaces. All our proofs are within Bishop’s informal system of constructive mathematics \(\mathrm {BISH}\).

References

  1. 1.
    Bishop, E.: Foundations of Constructive Analysis. McGraw-Hill, New York (1967)MATHGoogle Scholar
  2. 2.
    Bishop, E., Bridges, D.: Constructive Analysis. Grundlehren der Mathematischen Wissenschaften 279. Springer, Heidelberg (1985)MATHCrossRefGoogle Scholar
  3. 3.
    Bridges, D., Reeves, S.: Constructive mathematics in theory and programming practice. Philosophia Mathe. 3, 65–104 (1999)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bridges, D.S.: Reflections on function spaces. Ann. Pure Appl. Logic 163, 101–110 (2012)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Gillman, L., Jerison, M.: Rings of Continuous Functions. Van Nostrand, Princeton (1960)MATHCrossRefGoogle Scholar
  6. 6.
    Ishihara, H.: Relating bishop’s function spaces to neighborhood spaces. Ann. Pure Appl. Logic 164, 482–490 (2013)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Ishihara, H., Palmgren, E.: Quotient topologies in constructive set theory and type theory. Ann. Pure Appl. Logic 141, 257–265 (2006)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Petrakis, I.: Bishop spaces: constructive point-function topology. In: Mathematisches Forschungsinstitut Oberwolfach Report No. 52/2014, Mathematical Logic: Proof Theory, Constructive Mathematics, pp. 26–27. doi:10.4171/OWR/2014/52

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.University of MunichMunichGermany

Personalised recommendations