Journal of the Ramanujan Mathematical Society

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Hopfian and Co-Hopfian Zero-Dimensional Spaces


Affiliations
1 Department of Mathematics and Computer Science, R.D. University, Jabalpur-482001, India
2 Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta Canada T2N IN4, Canada
     

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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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  • Hopfian and Co-Hopfian Zero-Dimensional Spaces

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Authors

Satya Deo
Department of Mathematics and Computer Science, R.D. University, Jabalpur-482001, India
K. Varadarajan
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta Canada T2N IN4, Canada

Abstract


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.