Abstract
In this paper we show that some standard topological constructions may be fruitfully used in the theory of closure spaces (see [5], [4]). These possibilities are exemplified by the classical theorem on the universality of the Alexandroff's cube for T 0-closure spaces. It turns out that the closure space of all filters in the lattice of all subsets forms a “generalized Alexandroff's cube” that is universal for T 0-closure spaces. By this theorem we obtain the following characterization of the consequence operator of the classical logic: If ℒ is a countable set and C: P(ℒ) → P(ℒ) is a closure operator on X, then C satisfies the compactness theorem iff the closure space 〈ℒ,C〉 is homeomorphically embeddable in the closure space of the consequence operator of the classical logic.
We also prove that for every closure space X with a countable base such that the cardinality of X is not greater than 2ω there exists a subset X′ of irrationals and a subset X″ of the Cantor's set such that X is both a continuous image of X′ and a continuous image of X″.
We assume the reader is familiar with notions in [5].
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Engelking, General Topology, Monografie Matematyczne, PWN, Warszawa 1980.
C. C. Chang and H. J. Keisler, Model Theory, North-Holland, Amsterdam-London 1973.
A. W. Jankowski, An alternative characterisation of elementary logic, Bulletin de l'Academie Polonaise des Sciences, Series des Sciences Mathematiques, Vol. XXX, No. 1–2, 1982, pp. 9–13.
A. W. Jankowski, A characterization of the closed subsets of an 〈α, δ〉-closure space using an 〈α, δ〉-base, Bulletin de l'Academie Polonaise des Sciences, Series des Sciences Mathematiques, Vol. XXX, No 1–2, 1982, pp. 1–8.
A. W. Jankowski, A conjunction in closure spaces, Studia Logica, Vol. 43, No 4 (1984), pp. 341–351
H. Rasiowa and R. Sikorski, The Mathematics of Metamathematics, Monografie Matematyczne, PWN, Warszawa 1970.
H. Rasiowa, An Algebraic Approach to Non-Classical Logics, PWN-North-Holland, Warszawa-Amsterdam 1974.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Jankowski, A.W. Universality of the closure space of filters in the algebra of all subsets. Stud Logica 44, 1–9 (1985). https://doi.org/10.1007/BF00370806
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00370806