Abstract
Starting from the topological point of view a certain wide class of To-spaces is introduced having a very strong extension property for continuous functions with values in these spaces. It is then shown that all such spaces are complete lattices whose lattice structure determines the topology — these are the continuous lattices — and every such lattice has the extension property. With this foundation the lattices are studied in detail with respect to projections, subspaces, embeddings, and constructions such as products, sums, function spaces, and inverse limits. The main result of the paper is a proof that every topological space can be embedded in a continuous lattice which is homeomorphic (and isomorphic) to its own function space. The function algebra of such spaces provides mathematical models for the Church-Curry λ-calculus.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Literatur
J. C. Abbott, ed., Trends in Lattice Theory, Van Nostrand Reinhold Mathematical Studies, vol. 31 (1970).
P. Alexandroff and H. Hopf, Topologie I, Springer-Verlag, (1935).
E. Binz and H. H. Keller, Funktionenräume in der Kategorie der Limesräume, Annales Academiae Scientiarum Fennicae, Series A, I. Mathematica, no. 383 (1966), 21 pp.
G. Birkhoff, Lattice Theory, American Mathematical Society Colloquium Publications, vol. 25, Third (new) edition (1967).
E. Čech, Topological Spaces (revised by Z. Frolić and M. Katětov), Prague (1966).
G. Grätzer, Universal Algebra, Van Nostrand, (1968).
J. L. Kelley, General Topology, Van Nostrand, (1955).
E. Michael, Topologies on Spaces of Subsets, Transactions of the American Mathematical Society, vol. 77 (1951), pp. 152–182.
A. Nerode, Some Stone Spaces and Recursion Theory, Duke Mathematical Journal, vol. 26 (1959), pp. 397–406.
D. Scott, Outline of a Mathematical Theory of Computation, Proceedings of the Fourth Annual Princeton Conference on Information Sciences and Systems (1970), pp. 169–176.
_____, Lattice Theory, Data Types, and Semantices, New York University Symposia in Areas of Current Interest in Computer Science (Randall Rustin ed.) (1971) to appear.
_____, Lattice-theoretic Models for Various Type-free Calculi, Proceedings of the IVth International Congress for Logic, Methodology, and the Philosophy of Science, Bucharest (1972), to appear.
E. Spanier, Quasi-topologies, Duke Mathematical Journal, vol. 30 (1963) pp. 1–14.
W. Strother, Fixed Points, Fixed Sets, and M-Retracts, Duke Mathematical Journal, vol. 22 (1955), pp. 551–556.
Editor information
Rights and permissions
Copyright information
© 1972 Springer-Verlag Berlin · Heidelberg
About this paper
Cite this paper
Scott, D. (1972). Continuous lattices. In: Lawvere, F.W. (eds) Toposes, Algebraic Geometry and Logic. Lecture Notes in Mathematics, vol 274. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0073967
Download citation
DOI: https://doi.org/10.1007/BFb0073967
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-05920-2
Online ISBN: 978-3-540-37609-5
eBook Packages: Springer Book Archive