Failure of normality in the box product of uncountably many real lines

Lawrence, L.

doi:10.1090/S0002-9947-96-01375-X

Failure of normality in the box product of uncountably many real lines
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Trans. Amer. Math. Soc. 348 (1996), 187-203 Request permission

Abstract:

We prove in ZFC that the box product of $\omega _1$ many copies of $\omega +1$ is neither normal nor collectionwise Hausdorff. As an addendum to the proof, we show that if the cardinality of the continuum is $2^{\omega _1}$, then these properties also fail in the closed subspace consisting of all functions which assume the value $\omega$ on all but countably many indices.

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