An elementary theory of the category of topological spaces
HTML articles powered by AMS MathViewer
- by Dana I. Schlomiuk PDF
- Trans. Amer. Math. Soc. 149 (1970), 259-278 Request permission
Abstract:
An elementary system of axioms was given by F. W. Lawvere for the category of sets and mappings. The purpose of this paper is to provide a finite number of elementary axioms for the category of topological spaces and continuous mappings and to prove that any model of these axioms is equivalent to “the category of topological spaces” constructed over some model of Lawvere’s axioms. Furthermore, we prove that any complete category, model of the given axioms is equivalent to the category of topological spaces.References
- Richard F. Arens, A topology for spaces of transformations, Ann. of Math. (2) 47 (1946), 480–495. MR 17525, DOI 10.2307/1969087
- Samuel Eilenberg and Saunders MacLane, General theory of natural equivalences, Trans. Amer. Math. Soc. 58 (1945), 231–294. MR 13131, DOI 10.1090/S0002-9947-1945-0013131-6
- Ralph H. Fox, On topologies for function spaces, Bull. Amer. Math. Soc. 51 (1945), 429–432. MR 12224, DOI 10.1090/S0002-9904-1945-08370-0
- Peter Freyd, Abelian categories. An introduction to the theory of functors, Harper’s Series in Modern Mathematics, Harper & Row, Publishers, New York, 1964. MR 0166240
- J. R. Isbell, Subjects, adequacy, completeness and categories of algebras. [Subobjects, adequacy, completeness and categories of algebras], Rozprawy Mat. 36 (1964), 33. MR 163939
- John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144 F. W. Lawvere, Functorial semantics of algebraic theories, Dissertation, Columbia University, New York, 1963.
- F. William Lawvere, An elementary theory of the category of sets, Proc. Nat. Acad. Sci. U.S.A. 52 (1964), 1506–1511. MR 172807, DOI 10.1073/pnas.52.6.1506
- F. William Lawvere, The category of categories as a foundation for mathematics, Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) Springer, New York, 1966, pp. 1–20. MR 0207517
- Saunders Mac Lane, Categorical algebra, Bull. Amer. Math. Soc. 71 (1965), 40–106. MR 171826, DOI 10.1090/S0002-9904-1965-11234-4
- Barry Mitchell, Theory of categories, Pure and Applied Mathematics, Vol. XVII, Academic Press, New York-London, 1965. MR 0202787 D. Schlomiuk, A characterization of the category of topological spaces, Notices Amer. Math. Soc. 14 (1967), 269. Abstract #67T-180.
- E. Spanier, Quasi-topologies, Duke Math. J. 30 (1963), 1–14. MR 144300
Additional Information
- © Copyright 1970 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 149 (1970), 259-278
- MSC: Primary 18.10
- DOI: https://doi.org/10.1090/S0002-9947-1970-0258914-7
- MathSciNet review: 0258914