diff --git a/properties/P000207.md b/properties/P000207.md index ab46ec22c9..eb604f09c3 100644 --- a/properties/P000207.md +++ b/properties/P000207.md @@ -3,18 +3,25 @@ uid: P000207 name: Strongly collectionwise normal aliases: - Divisible +- Almost $2$-fully normal refs: - zb: "0046.16403" name: Sur une problème de M. Dieudonné (Cohen) - doi: 10.1090/S0002-9939-1981-0630058-4 name: Strong collectionwise normality and M. E. Rudin’s Dowker space (Hart) + - doi: 10.2307/1993026 + name: Some Generalizations of Full Normality (M. J. Mansfield) --- For each neighborhood $U$ of the diagonal $\Delta=\{(x,x)\mid x\in X\}$ in $X\times X$, there is a neighborhood $V$ of the diagonal such that $V\circ V\subseteq U$. -Studied by Cohen in {{zb:0046.16403}} +In Theorem 2.6 of {{doi:10.2307/1993026}} this property was shown to be equivalent to +almost $2$-fully normal: each open cover $\mathcal U$ has an open almost $2$-star +refinement $\mathcal V$, that is $\mathcal{V}$ is a refinement of $\mathcal{U}$ and for any $x, y, z$ with $y, z\in \text{St}(x, \mathcal{V})$ there exists $U\in\mathcal{U}$ with $y, z\in U$. + +Originally studied by Cohen in {{zb:0046.16403}} (), where it was shown to be strictly stronger than {P88}. diff --git a/spaces/S000138/properties/P000207.md b/spaces/S000138/properties/P000207.md index 010d5632db..3e11dbc4e9 100644 --- a/spaces/S000138/properties/P000207.md +++ b/spaces/S000138/properties/P000207.md @@ -5,8 +5,7 @@ value: true refs: - doi: 10.1090/S0002-9939-1982-0671226-6 name: More on M. E. Rudin’s Dowker space (K. P. Hart) -- doi: 10.2307/1993026 - name: Some Generalizations of Full Normality (M. J. Mansfield) --- -By theorem 2.2 of {{doi:10.1090/S0002-9939-1982-0671226-6}}, {S138} is $2$-fully normal, and so almost $2$-fully normal. By theorem 2.6 of {{doi:10.2307/1993026}}, it's strongly collectionwise normal. +By theorem 2.2 of {{doi:10.1090/S0002-9939-1982-0671226-6}}, +{S138} is $2$-fully normal, and so {{P000207}}.