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A Note on Inverse Limits of Finite Spaces

Published online by Cambridge University Press:  20 November 2018

S. B. Nadler Jr
S. B. Nadler Jr*
Affiliation:
Dalhousie University, Halifax, Nova Scotia
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The following lemma, which appears as Lemma 4 in [5], was used to determine certain multicoherence properties of inverse limits of continua.

Lemma. Let X denote the inverse limit of an inverse system {Xλ, fλμ, Λ} of compact Hausdorff spaces Xλ. If Xλ has no more than k components (where k < ∞ is fixed) for each λ ∊ Λ, then X has no more than k components.

In this paper we give a set theoretic analogue of this lemma and an extension which was suggested to the author by Professor F. W. Lawvere. An application to inverse limits of finite groups is then given.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 1 , March 1970 , pp. 69 - 70
Copyright
Copyright © Canadian Mathematical Society 1970

Footnotes

(1)

Part of this work was supported by National Research Grant A7346.

References

1. Capel, C. E., Inverse limit spaces, Duke Math. J. 21 (1954), 233-245.Google Scholar
2. Eilenberg, S. and Steenrod, N., Foundations of Algebraic Topology, Princeton Univ. Press, Princeton, N.J., 1952.Google Scholar
3. Hocking, J. G. and Young, G. S., Topology, Addison-Wesley, Reading, Mass., 1961.Google Scholar
4. Kelley, J. L., General Topology, Van Nostrand, Princeton, N.J., 1955.Google Scholar
5. Nadler, S. B. Jr., Inverse limits, multicoherence, and hyperspaces (to appear).Google Scholar