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Notes on general topology

III. A non-metric image of an ordered compactum

Published online by Cambridge University Press:  24 October 2008

A. J. Ward
A. J. Ward
Affiliation:
Emmanuel College, Cambridge
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Extract

It has been known for some time that the product of a non-metrizable Hausdorff space and any (non-trivial) Hausdorff space cannot be the continuous image of an ordered continuum. (For a survey of this and related problems, see Mardešić and Papić ((1)).) Further, it has been shown by Treybig ((2)) (and independently by the present author) that the product of two Hausdorff spaces cannot even be the continuous image of an ordered compactum unless both the spaces are metrizable or one is finite. It is therefore of some interest to give a simple example of a space X which is the continuous image of an ordered compactum K and contains the product of a non-metrizable space and an infinite discrete space, imbedded in such a way as to form a sequence of homeomorphic subsets with a connected (non-trivial) topological limit.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 61 , Issue 4 , October 1965 , pp. 881 - 882
Copyright
Copyright © Cambridge Philosophical Society 1965

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References

REFERENCES

(1) Mardešić, S. and Papić, P. Neki problemi preslikavarja uredjenik kompakta (in Serbo-Croat; summary in English, and bibliography), in Neki nernšeni problemi u matematici, Mat. biblioteka 25 (1963). A bibliography will also be found in the same authors' paper Continuous images of ordered compacta.… Glasnik Mat.-Fiz. Astronom. Društvo Mat. Fiz. Hrvatske Ser. II 17 (1962), 3–25, which may be more accessible.Google Scholar
(2) Treybig, L. B. Concerning continuous images of compact ordered spaces. Proc. American Math. Soc. 15 (1964), 866–871.CrossRefGoogle Scholar