add almost 2 fully normal characterization to strongly collectionwise normal

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}}
 (<http://gallica.bnf.fr/ark:/12148/bpt6k3186w/f290.image>),
 where it was shown to be
 strictly stronger than {P88}.

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}}.