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On the number of topologies definable for a finite set

Published online by Cambridge University Press:  09 April 2009

A. Shafaat
A. Shafaat
Affiliation:
Institute of Advanced Studies The Australian National UniversityCanberra, A.C.T. and Panjab University Lahore, Pakistan
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No general rule for determining the number N(n) of topologies definable for a finite set of cardinal n is known. In this note we relate N(n) to a function Ft(r1,…, rt+1) defined below which has a simple combinatorial interpretation. This relationship seems useful for the study of N (n). In particular this can be used to calculate N(n) for small values. For n 3, 4, 5, 6 we find N(3) = 29, N(4) = 355, N(5) = 7,181, N(6) = 145,807.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 8 , Issue 2 , May 1968 , pp. 194 - 198
Copyright
Copyright © Australian Mathematical Society 1968

References

1 Strictly speaking, members of B 4-i must be taken as the unions ∪ Y(i, k) of all sets represented by the x's in Y(i, k), but since x's represent disjoint sets this will not effect our conclusion about F 3(r 1 …, r 4)