Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-26T08:59:55.240Z Has data issue: false hasContentIssue false
  • English
  • Français

Neighbourhood lattices – a poset approach to topological spaces

Published online by Cambridge University Press:  17 April 2009

Frank P. Prokop
Frank P. Prokop
Affiliation:
Department of Mathematics, The University of Wollongong, Wollongong, N.S.W., Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper neighbourhood lattices are developed as a generalisation of topological spaces in order to examine to what extent the concepts of “openness”, “closedness”, and “continuity” defined in topological spaces depend on the lattice structure of P(X), the power set of X.

A general pre-neighbourhood system, which satisfies the poset analogues of the neighbourhood system of points in a topological space, is defined on an ∧-semi-lattice, and is used to define open elements. Neighbourhood systems, which satisfy the poset analogues of the neighbourhood system of sets in a topological space, are introduced and it is shown that it is the conditionally complete atomistic structure of P(X) which determines the extension of pre-neighbourhoods of points to the neighbourhoods of sets.

The duals of pre-neighbourhood systems are used to generate closed elements in an arbitrary lattice, independently of closure operators or complementation. These dual systems then form the backdrop for a brief discussion of the relationship between preneighbourhood systems, topological closure operators, algebraic closure operators, and Čech closure operators.

Continuity is defined for functions between neighbourhood lattices, and it is proved that a function f: X → Y between topological spaces is continuous if and only if corresponding direct image function between the neighbourhood lattices P(X) and P(Y) is continuous in the neighbourhood sense. Further, it is shown that the algebraic character of continuity, that is, the non-convergence aspects, depends only on the properites of pre-neighbourhood systems. This observation leads to a discussion of the continuity properties of residuated mappings. Finally, the topological properties of normality and regularity are characterised in terms of the continuity properties of the closure operator on a topological space.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 39 , Issue 1 , February 1989 , pp. 31 - 48
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Abbott, J.C., ‘Remarks on an Algebraic structure for a Topology’, Proc. 2nd Prague Topological Symposium (1966), 17–21.Google Scholar
[2]Birkhoff, G., Lattice Theory (A.M.S., Providence, 1967).Google Scholar
[3]Blythe, T.S. and Janowitz, M.F., Residuation Theory (Pergammon Press, Oxford, 1972).Google Scholar
[4]Čech, E., Topological Spaces, Vols I and II (Interscience Publishers, London, N.Y., Sydney, 1966).Google Scholar
[5]Dowker, C.H. and Papert, D., ‘Quotient frames and subspaces’, Proc. London Math. Soc. 10 (1966), 275–296.CrossRefGoogle Scholar
[6]Dowker, C.H. and Strauss, D., ‘Separation axions for frames’, Colloq. Math. Soc. János Bolyai 8 (1972), 223–240.Google Scholar
[7]Gratzer, G., Lattice Theory: First Concepts and Distributive Lattices (Freeman, San Francisco, 1971).Google Scholar
[8]Isbell, J.R., ‘Atomless parts of spaces’, Math. Scand. 31 (1972), 5–32.Google Scholar
[9]Johnstone, P.T., ‘The point of pointless topology’, Bull. Amer. Math. Soc. 8 (1983), 41–53.CrossRefGoogle Scholar
[10]Jónsson, B. and Tarski, A., ‘Boolean algebras with operators’, Amer. J. Math. 73 (1951), 891–939.CrossRefGoogle Scholar
[11]Kelley, J.L., General Topology (Van Nostrand, Princeton, N.J., 1955).Google Scholar
[12]MacLane, S. and Birkhoff, G., Algebra (Macmillan, N.Y., 1966).Google Scholar
[13]McKinsey, J.C.C. and Tarski, A., ‘The algebra of topology’, Ann. of Math (2)45 (1944), 141–191.Google Scholar
[14]McKinsey, J.C.C. and Tarski, A., ‘On closed elements in closure algebras’, Ann. of Math.(2) 47 (1946), 122–162.Google Scholar
[15]Prokop, F.P., Pre-neighbourhood ∧-semilattices, neighbourhood lattices and their associated continuous functions (Ph.D. thesis, Univ. of Wollongong, Wollougong, N.S.W., 1980).Google Scholar
[16]Tukey, J.W., Convergence and Uniformity in Topology (Princeton Univ. Press, Princeton, N.J., 1940).Google Scholar