small update

properties/P000193.md

@@ -10,4 +10,4 @@ refs:
  name: Shrinking space
 ---
 
-A space in which every open cover admits a shrinking; that is, a space 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 $U \in \mathscr U$, $\mathrm{cl}_X s(U) \subseteq U$.
+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 $U \in \mathscr U$, $\mathrm{cl}_X s(U) \subseteq U$.

properties/P000194.md

@@ -8,6 +8,8 @@ refs:
  name: Characterizations of developable topological spaces (J. Worrell and H. Wicke)
 - zb: "0413.54027"
  name: On Submetacompactness (H. Junnila)
+- zb: "0712.54016"
+ name: Generalized paracompactness (Y. Yasui)
 ---
 
-A space in which every open cover has a $\theta$-sequence of open refinements; that is, a space in which, for every open cover $\mathscr U$, there exists a sequence $\langle \mathscr V_n : n \in \omega\rangle$ of open covers where each $\mathscr V_n$ is a refinement of $\mathscr U$ and, for each point $x$ of the space, there exists $n \in \omega$ such that $\mathscr V_n$ is point-finite at $x$.
+A space in which every open cover has a $\theta$-sequence of open refinements; that is, a space $X$ in which, for every open cover $\mathscr U$, there exists a sequence $\langle \mathscr V_n : n \in \omega\rangle$ of open covers where each $\mathscr V_n$ is a refinement of $\mathscr U$ and, for each point $x \in X$, there exists $n \in \omega$ such that $\mathscr V_n$ is point-finite at $x$.

theorems/T000541.md

@@ -0,0 +1,12 @@
+---
+uid: T000541
+if:
+ P000193: true
+then:
+ P000013: 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 $E \subseteq X \semtinus \mathrm{cl}_X(U)$, $F \subseteq X \semtminus \mathrm{cl}_X(V)$, and $\left( X \setminus \mathrm{cl}_X(U) \right) \cap \left( X \setminus \mathrm{cl}_X(V) \right) = X \setminus \left( \mathrm{cl}_X(U) \cup \mathrm{cl}_X(V) \right) = \emptyset$.