Set up definitions

properties/P000193.md

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+---
+uid: P000193
+name: Shrinking
+refs:
+- zb: "1059.54001"
+ name: Encyclopedia of general topology
+- zb: "0712.54016"
+ name: Generalized paracompactness (Y. Yasui)
+- wikipedia: Shrinking_space
+ 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 $U \in \mathscr U$, $\mathrm{cl}_X s(U) \subseteq U$.

properties/P000194.md

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+---
+uid: P000194
+name: Submetacompact
+aliases:
+- $\theta$-refinable
+refs:
+- zb: "0132.18401"
+ name: Characterizations of developable topological spaces (J. Worrell and H. Wicke)
+- zb: "0413.54027"
+ name: On Submetacompactness (H. Junnila)
+---
+
+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 $x$.