countably paracompact update

properties/P000193.md

@@ -12,4 +12,4 @@ refs:
 
 A space in which every open cover admits a shrinking; that is, a space $X$ in which, given any open cover $\mathscr U$, there is a function $s : \mathscr U \to \tau_X \setminus \{\emptyset\}$ such that $s[\mathscr U]$ is an open cover and, for each $U \in \mathscr U$, $\mathrm{cl}_X s(U) \subseteq U$.
 
-See also [Dan Ma's Topology Blog post on spaces with shrinking properties](https://dantopology.wordpress.com/2017/01/05/spaces-with-shrinking-properties/).
+See also [Dan Ma's Topology Blog post on Spaces with shrinking properties](https://dantopology.wordpress.com/2017/01/05/spaces-with-shrinking-properties/).

theorems/T000541.md

@@ -3,12 +3,16 @@ uid: T000541
 if:
  P000193: true
 then:
- P000013: true
+ and:
+ - P000013: true
+ - P000032:true
 refs:
 - zb: "0712.54016"
  name: Generalized paracompactness (Y. Yasui)
 ---
 
-The shrinking property is motivated as a generalization of {P13} in {{zb:0712.54016}}. The argument there goes as follows. Suppose $E$ and $F$ are disjoint closed subsets of a shrinking space $X$. Then $\{ X \setminus E , X \setminus F\}$ is an open cover of $X$, so there exists an open cover $\{U, V\}$ of $X$ such that $\mathrm{cl}_X(U) \subseteq X \setminus E$ and $\mathrm{cl}_X(V) \subseteq X \setminus F$. Note then that $E \subseteq X \setminus \mathrm{cl}_X(U)$, $F \subseteq X \setminus \mathrm{cl}_X(V)$, and $\left( X \setminus \mathrm{cl}_X(U) \right) \cap \left( X \setminus \mathrm{cl}_X(V) \right) = X \setminus \left( \mathrm{cl}_X(U) \cup \mathrm{cl}_X(V) \right) = \emptyset$.
+The shrinking property is motivated as a generalization of {P13} in {{zb:0712.54016}}. The argument there for {P193} implies {P13} goes as follows. Suppose $E$ and $F$ are disjoint closed subsets of a shrinking space $X$. Then $\{ X \setminus E , X \setminus F\}$ is an open cover of $X$, so there exists an open cover $\{U, V\}$ of $X$ such that $\mathrm{cl}_X(U) \subseteq X \setminus E$ and $\mathrm{cl}_X(V) \subseteq X \setminus F$. Note then that $E \subseteq X \setminus \mathrm{cl}_X(U)$, $F \subseteq X \setminus \mathrm{cl}_X(V)$, and $\left( X \setminus \mathrm{cl}_X(U) \right) \cap \left( X \setminus \mathrm{cl}_X(V) \right) = X \setminus \left( \mathrm{cl}_X(U) \cup \mathrm{cl}_X(V) \right) = \emptyset$.
 
-See also [Dan Ma's Topology Blog post on spaces with shrinking properties](https://dantopology.wordpress.com/2017/01/05/spaces-with-shrinking-properties/).
+See also [Dan Ma's Topology Blog post on Spaces with shrinking properties](https://dantopology.wordpress.com/2017/01/05/spaces-with-shrinking-properties/).
+
+The implication that {P193} implies {P32} appears in the diagram on page 191 of {{zb:0712.54016}} and is mentioned in passing in {{zb:1059.54001}} on page 199. See also Theorem 5 at [Dan Ma's Topology Blog post on Countably paracompact spaces](https://dantopology.wordpress.com/2016/12/08/countably-paracompact-spaces/) for more.