Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-28T13:40:00.646Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Banach, S.Un théorème sur les transformations biunivoques. Fund. Math. 6 (1924), 236–239.Google Scholar
(2)Bruijn, N. G. de, Gemeenschappelijke representantensystemen van twee Klassindeelingen van een verzameling. Nieuw Arch. Wisk. (2) 22 (1943), 48–52.Google Scholar
(3)Hall, Marshall, Distinct representatives of subsets. Bull. Amer. Math. Soc. 54 (1948), 922–926.Google Scholar
(4)Hall, P.On representatives of subsets. J. London Math. Soc. 10 (1935), 26–30.CrossRefGoogle Scholar
(5)Ore, O.Graphs and matching theorem. Duke Math. J. 22 (1955), 625–639.Google Scholar