Shrinking property (#782)

properties/P000193.md

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+---
+uid: P000193
+name: Shrinking
+aliases:
+- Has the shrinking property
+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 $X$ in which, given any open cover $\{ U_\alpha : \alpha \in A\}$, there is an open cover $\{ V_\alpha : \alpha \in A\}$ such that $\overline{V_\alpha} \subseteq U_\alpha$ for each $\alpha \in A$.
+
+See also [Dan Ma's Topology Blog post on Spaces with shrinking properties](https://dantopology.wordpress.com/2017/01/05/spaces-with-shrinking-properties/).

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)
+- 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 $X$ 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 $x$.
+
+This property was introduced in {{zb:0132.18401}} under the name of *$\theta$-refinable*, and later renamed to *submetacompact* in {{zb:0413.54027}} (full article available [here](http://topology.nipissingu.ca/tp/reprints/v03/tp03207s.pdf)).

theorems/T000542.md

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+---
+uid: T000542
+if:
+ P000193: true
+then:
+ P000013: true
+
+refs:
+- zb: "0712.54016"
+ name: Generalized paracompactness (Y. Yasui)
+---
+
+The argument in {{zb:0712.54016}} for this result 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 $\overline{U} \subseteq X \setminus E$ and $\overline{V} \subseteq X \setminus F$. Note then that $E \subseteq X \setminus \overline{U}$, $F \subseteq X \setminus \overline{V}$, and $\left( X \setminus \overline{U} \right) \cap \left( X \setminus \overline{V} \right) = X \setminus \left( \overline{U} \cup \overline{V} \right) = \emptyset$.
+
+See also [Dan Ma's Topology Blog post on Spaces with shrinking properties](https://dantopology.wordpress.com/2017/01/05/spaces-with-shrinking-properties/).

theorems/T000543.md

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+---
+uid: T000543
+if:
+ P000193: true
+then:
+ P000032: true
+refs:
+- zb: "0712.54016"
+ name: Generalized paracompactness (Y. Yasui)
+- zb: "1059.54001"
+ name: Encyclopedia of general topology
+- zb: "1052.54001"
+ name: General Topology (S. Willard)
+---
+
+This implication 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 21.3 of {{zb:1052.54001}} and Theorem 5 at [Dan Ma's Topology Blog post on Countably paracompact spaces](https://dantopology.wordpress.com/2016/12/08/countably-paracompact-spaces/).

theorems/T000544.md

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+---
+uid: T000544
+if:
+ P000031: true
+then:
+ P000194: true
+refs:
+- zb: "0413.54027"
+ name: On Submetacompactness (H. Junnila)
+---
+
+This is evident from the definitions. The single point-finite open refinement guaranteed by metacompactness generates a sequence with the desired properties.

theorems/T000545.md

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+---
+uid: T000545
+if:
+ and:
+ - P000194: true
+ - P000013: true
+then:
+ P000193: true
+refs:
+- zb: "0712.54016"
+ name: Generalized paracompactness (Y. Yasui)
+---
+
+See Theorem 6.2 of {{zb:0712.54016}}.