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An extension of Banach's mapping theorem, with applications to problems concerning common representatives

Published online by Cambridge University Press:  24 October 2008

Hazel Perfect  and
J. S. Pym
Hazel Perfect
Affiliation:
Department of Pure Mathematics, University of Sheffield
J. S. Pym
Affiliation:
Department of Pure Mathematics, University of Sheffield
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Extract

In this paper, we give an extension (Theorem 1) of the following well-known result of Banach ((l)): If X, Y are sets and Θ: X → Y, ψ: Y → X are injective mappings, then there exist partitions X = X1 ∪ X2, Y = Y1 ∪ Y2 such that Θ(X1) = Y1 and ψ(Y2) = X2. Here, as is usual, we say that X = X1 ∪ X2 is a partition of the set X = X1 ∩ X2 = π. Our theorem is applied in sections 2 and 3 to problems concerned with the existence of common representatives for two families of sets.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 62 , Issue 2 , April 1966 , pp. 187 - 192
Copyright
Copyright © Cambridge Philosophical Society 1966

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References

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