Regular matrices and 𝑃-sets in 𝛽𝑁\𝑁

Atalla, R. E.

doi:10.1090/S0002-9939-1973-0324655-9

Regular matrices and $P$-sets in $\beta N\backslash N$
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by R. E. Atalla

Proc. Amer. Math. Soc. 37 (1973), 157-162

DOI: https://doi.org/10.1090/S0002-9939-1973-0324655-9

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Abstract:

A P-set is a closed set which is interior to any zero set (closed ${G_\delta }$) which contains it. Henriksen and Isbell showed that the ’support set’ in $\beta N\backslash N$ of a nonnegative regular matrix is a P-set. We show that each such support set contains a family of ${2^c}$ pairwise disjoint perfect nowhere dense P-sets, so that not every P-set comes from a matrix. Moreover, each of the P-sets produced is the support of a Borel probability measure on $\beta N\backslash N$.

References

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