Subcompact property

properties/P000234.md

@@ -0,0 +1,14 @@
+---
+uid: P000234
+name: Subcompact
+---
+
+*from discussion at STDC, will add refs later*
+
+There exists a basis $\mathcal B$ of nonempty sets
+such that for every regular filter base
+$\mathcal F\subseteq\mathcal B$, $\bigcap\mathcal F\not=\emptyset$.
+
+A regular filter base $\mathcal F$ has the property that for each
+$A,B\in\mathcal F$, there exists $C\in\mathcal F$ such that
+$\operatorname{cl}(C)\subseteq A\cap B$. 

properties/P000235.md

@@ -0,0 +1,6 @@
+---
+uid: P000235
+name: Complete pair based
+---
+
+*from discussion at STDC, will add refs later*

properties/P000236.md

@@ -0,0 +1,6 @@
+---
+uid: P000236
+name: Generalized subcompact
+---
+
+*from discussion at STDC, will add refs later*

theorems/T000645.md

@@ -2,7 +2,7 @@
 uid: T000645
 if:
   and:
-    - P000063: true
+    - P000236: true
     - P000137: false
 then:
   P000206: true
@@ -13,6 +13,8 @@ refs:
     name: Classical Descriptive Set Theory (Kechris)
 ---
 
+*TODO*
+
 For a {P3} {P16} space, Player 2 may win by choosing $V_n$
 with $\overline{V_n}\subseteq U_n$.
 

theorems/T000680.md

@@ -7,9 +7,11 @@ if:
   - P000137: false
 
 then:
-  P000206: true
+  P000234: true
 ---
 
+TODO update:
+
 A space $X$ is {P206} if and only if the Kolmogorov quotient of $X$ is {P206}.
 
 The Kolmogorov quotient of $X$ is {P23}, {P3} and non-empty, so strongly Choquet [(Explore)](https://topology.pi-base.org/spaces?q=weakly+locally+compact+%2B+T_2+%2B+not+empty+%2B+not+strongly+choquet).

theorems/T000862.md

@@ -0,0 +1,9 @@
+---
+uid: T000862
+if:
+  P000234: true
+then:
+  P000235: true
+---
+
+Immediate from Lynne Yengulalp.

theorems/T000863.md

@@ -0,0 +1,9 @@
+---
+uid: T000863
+if:
+  P000055: true
+then:
+  P000234: true
+---
+
+Immediate from Lynne Yengulalp.

theorems/T000864.md

@@ -0,0 +1,9 @@
+---
+uid: T000864
+if:
+  P000235: true
+then:
+  P000236: true
+---
+
+Immediate from Lynne Yengulalp.

theorems/T000865.md

@@ -0,0 +1,9 @@
+---
+uid: T000865
+if:
+  P000063: true
+then:
+  P000235: true
+---
+
+Immediate from Lynne Yengulalp.