Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-11T01:24:05.324Z Has data issue: false hasContentIssue false
  • English
  • Français

Functional equations in total negation

Part of: Basic constructions Fairly general properties Generalities

Published online by Cambridge University Press:  09 April 2009

T. B. M. McMaster  and
C. R. Turner
T. B. M. McMaster
Affiliation:
Pure Mathematics Department Queen's UniversityBelfast BT7 1NN NorthernIreland e-mail: t.b.m.mcmaster@qub.ac.uk
C. R. Turner
Affiliation:
School of Electrical and Mechanical Engineering University of Ulster at JordanstownNorthernIreland e-mail: c.turner@ulst.ac.uk
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.

It is known that the only topological invariants P for which anti(P) = anti2 (P), anti( ) denoting Bankston's total negation operator, are those which are determined purely by the cardinality of the underlying point-set. We examine equations of the form antin (P) = antin (not P), reaching similar conclusions for n ≤ 2 but weaker ones for n > 3. A corresponding investigation for total negation within a constraint is initiated.

MSC classification

Secondary: 54A25: Cardinality properties (cardinal functions and inequalities, discrete subsets) 54B05: Subspaces 54D99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 68 , Issue 3 , June 2000 , pp. 334 - 339
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Bankston, P., ‘The total negation of a topological property’, Illinois J. Math. 23 (1979), 241–252.CrossRefGoogle Scholar
[2]Levine, N., ‘On the equivalence of compactness and finiteness in topology’, Amer. Math. Monthly 75 (1968), 178–180.CrossRefGoogle Scholar
[3]Matier, J. and McMaster, T. B. M., ‘Total negation in general topology’, Irish Math. Soc. Bull. 25 (1990), 26–37.CrossRefGoogle Scholar
[4]Matier, J. and Mcmaster, T. B. M., ‘Iteration of Bankston's ‘anti’-operation’, J. Inst. Math. and Comp. Sci. Math. Ser. 3 (1990), 31–35.Google Scholar
[5]Matier, J. and McMaster, T. B. M., ‘Total negation of separability and related properties’, Proc. Roy. Irish Acad. Sec. A 90 (1990), 131–137.Google Scholar
[6]Matthews, P. T. and McMaster, T. B. M., ‘Families of spaces having prescribed embeddability order-type’, Rend. Istit. Mat. Univ. Trieste 25 (1993), 345–352.Google Scholar
[7]Matthews, P. T. and McMaster, T. B. M., ‘Minimal spaces, maximal pre-antis’, Rend. Istit. Mat. Univ. Trieste 25 (1993), 353–361.Google Scholar
[8]McMaster, T. B. M. and Turner, C. R., ‘Constrained total negation and its iteration’, preprint (Queen's University of Belfast, 1998).Google Scholar
[9]McMaster, T. B. M. and Turner, C. R., ‘Total negation under constraint: pre-anti properties’, preprint (Queen's University of Belfast, 1998).Google Scholar
[10]Reilly, I. L. and Vamanamurthy, M. K., ‘Compactness and finiteness’, Colloq. Math. Soc. János Bolyai 23 (1978), 1033–1041.Google Scholar
[11]Reilly, I. L. and Vamanamurthy, M. K., ‘Some topological anti-properties’, Illinois J. Math 24 (1980), 382–389.CrossRefGoogle Scholar
[12]Turner, C. R., Total negation of topological properties in constrained environments (Ph.D. Thesis, Queen's University of Belfast, 1997).Google Scholar
[13]Wilansky, A., ‘Between T 1 and T 2‘, Amer. Math. Monthly 74 (1967), 261–266.Google Scholar