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

  • K. D. Magill (a1)

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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)).

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References

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1.COOK, H. and INGRAM, W. T., Obtaining AR-like continua as inverse limits with only two bonding maps, Glas. Mat. Ser. III 4 (1969), 309312.
2.HUREWICZ, W. and WALLMAN, H., Dimension Theory (Princeton University Press, Princeton, 1941).
3.Magill, K. D. Jr, Some Open Problems and Directions for Further Research in Semigroups of Continuous Selfmaps (Univ. Alg. and Apps., Banach Center Pub., PWN-Polish Sci. Pub., Warsaw 9, 1982), 439454.
4.Michael, E. A., On a theorem of Rudin and Klee, Proc. Amer. Math. Soc. 12 (1961), 921.
5.Subbiah, S., Some finitely generated subsemigroups of S(X), Fund. Math. 86 (1975), 221231.
6.Subbiah, S., The compact-open topology for semigroups of continuous selfmaps, Semigroup Forum 35 (1987), 2933

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