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More On the countability index and the density index of S(X)

Published online by Cambridge University Press:  20 January 2009

K. D. Magill Jr
K. D. Magill Jr
Affiliation:
106 Dlefendorf Hall, Suny at Buffalo Buffalo, New York 14214-3093, U.S.A.
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The countability index, C(S), of a semigroup S is the smallest integer n, if it exists, such that every countable subset of S is contained in a subsemigroup with n generators. If no such integer exists, define C(S) = ∞. The density index, D(S), of a topological semigroup S is the smallest integer n, if it exists, such that S contains a dense subsemigroup with n generators. If no such integer exists, define D(S) = ∞. S(X) is the topological semigroup of all continuous selfmaps of the locally compact Hausdorff space X where S(X) is given the compact-open topology. Various results are obtained about C(S(X)) and D(S(X)). For example, if X consists of a finite number (< 1) of components, each of which is a compact N-dimensional subspace of Euclidean Nspace and has the internal extension property and X is not the two point discrete space. Then C(S(X)) exceeds two but is finite. There are additional results for C(S(X)) and similar results for D(S(X)).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 33 , Issue 1 , February 1990 , pp. 159 - 164
Copyright
Copyright © Edinburgh Mathematical Society 1990

References

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