From 4f0b48728de782070b1331a8e49b1f5fbdf0d4d1 Mon Sep 17 00:00:00 2001 From: Chris <30360237+ccaruvana@users.noreply.github.com> Date: Mon, 7 Oct 2024 19:45:33 -0400 Subject: [PATCH] Shrinking property (#782) --- properties/P000193.md | 17 +++++++++++++++++ properties/P000194.md | 17 +++++++++++++++++ theorems/T000542.md | 15 +++++++++++++++ theorems/T000543.md | 16 ++++++++++++++++ theorems/T000544.md | 12 ++++++++++++ theorems/T000545.md | 14 ++++++++++++++ 6 files changed, 91 insertions(+) create mode 100644 properties/P000193.md create mode 100644 properties/P000194.md create mode 100644 theorems/T000542.md create mode 100644 theorems/T000543.md create mode 100644 theorems/T000544.md create mode 100644 theorems/T000545.md diff --git a/properties/P000193.md b/properties/P000193.md new file mode 100644 index 000000000..51456cc4d --- /dev/null +++ b/properties/P000193.md @@ -0,0 +1,17 @@ +--- +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/). \ No newline at end of file diff --git a/properties/P000194.md b/properties/P000194.md new file mode 100644 index 000000000..7b4f3d506 --- /dev/null +++ b/properties/P000194.md @@ -0,0 +1,17 @@ +--- +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)). diff --git a/theorems/T000542.md b/theorems/T000542.md new file mode 100644 index 000000000..b17d3f2f0 --- /dev/null +++ b/theorems/T000542.md @@ -0,0 +1,15 @@ +--- +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/). \ No newline at end of file diff --git a/theorems/T000543.md b/theorems/T000543.md new file mode 100644 index 000000000..6a908479e --- /dev/null +++ b/theorems/T000543.md @@ -0,0 +1,16 @@ +--- +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/). \ No newline at end of file diff --git a/theorems/T000544.md b/theorems/T000544.md new file mode 100644 index 000000000..9b1d80e43 --- /dev/null +++ b/theorems/T000544.md @@ -0,0 +1,12 @@ +--- +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. \ No newline at end of file diff --git a/theorems/T000545.md b/theorems/T000545.md new file mode 100644 index 000000000..5e1fb538a --- /dev/null +++ b/theorems/T000545.md @@ -0,0 +1,14 @@ +--- +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}}. \ No newline at end of file