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Hopfian and Co-Hopfian Zero-Dimensional Spaces
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The main results proved in this paper are:
(i) The only Hopfian or co-Hopfian objects among compact totally disconnected metrizable spaces are finite discrete spaces.
(ii) Every infinite closed subspace of (IN and hence any infinite closed subspace of N* = βN- N is non-co-Hopfian. However, N* admits an abundance of non-closed subspaces which are simultaneously Hopfian and co-Hopfian.
(iii) There are at least 2C non-homeomorphic compact totally disconnected perfect co-Hopfian spaces which are not rigid.
Keywords
Rigid Spaces, Rigid Spaces for a Class of Maps, Hopfian and Co-Hopfian Spaces, Continuous Displacements. F-Spaccs, Growth of a Space, Extremally Disconnected Spaces.
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