The intersection multiplicity of compact 𝑛-dimensional metric spaces

Weller, Glenn P.

doi:10.1090/S0002-9939-1972-0307194-X

The intersection multiplicity of compact $n$-dimensional metric spaces
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by Glenn P. Weller

Proc. Amer. Math. Soc. 36 (1972), 293-294

DOI: https://doi.org/10.1090/S0002-9939-1972-0307194-X

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Abstract:

It is shown that there is an integer $\mu (n)$ such that any compact n-dimensional metric space M has intersection multiplicity at most $\mu (n)$. That is, if $\mathcal {U}$ is an open cover of M, then there is an open cover $\mathcal {V}$ refining $\mathcal {U}$ such that any element of $\mathcal {V}$ can intersect at most $\mu (n)$ other elements of $\mathcal {V}$.

References

J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees. MR 0248844
John G. Hocking and Gail S. Young, Topology, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1961. MR 0125557
Yoshihiro Shikata, On the smoothing problem and the size of a topological manifold, Osaka Math. J. 3 (1966), 293–301. MR 215307