diff --git a/properties/P000234.md b/properties/P000234.md new file mode 100644 index 000000000..19bd416b3 --- /dev/null +++ b/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 +$\operatorname{cl}(C)\subseteq A\cap B$. diff --git a/properties/P000235.md b/properties/P000235.md new file mode 100644 index 000000000..82ff7288d --- /dev/null +++ b/properties/P000235.md @@ -0,0 +1,6 @@ +--- +uid: P000235 +name: Complete pair based +--- + +*from discussion at STDC, will add refs later* diff --git a/properties/P000236.md b/properties/P000236.md new file mode 100644 index 000000000..9ffea9805 --- /dev/null +++ b/properties/P000236.md @@ -0,0 +1,6 @@ +--- +uid: P000236 +name: Generalized subcompact +--- + +*from discussion at STDC, will add refs later* diff --git a/theorems/T000645.md b/theorems/T000645.md index 2b7bc8097..f52fa0d87 100644 --- a/theorems/T000645.md +++ b/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 $\overline{V_n}\subseteq U_n$. diff --git a/theorems/T000680.md b/theorems/T000680.md index 0b43a8ea2..6aca1ee3e 100644 --- a/theorems/T000680.md +++ b/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). diff --git a/theorems/T000862.md b/theorems/T000862.md new file mode 100644 index 000000000..a00117750 --- /dev/null +++ b/theorems/T000862.md @@ -0,0 +1,9 @@ +--- +uid: T000862 +if: + P000234: true +then: + P000235: true +--- + +Immediate from Lynne Yengulalp. diff --git a/theorems/T000863.md b/theorems/T000863.md new file mode 100644 index 000000000..5370f0cdf --- /dev/null +++ b/theorems/T000863.md @@ -0,0 +1,9 @@ +--- +uid: T000863 +if: + P000055: true +then: + P000234: true +--- + +Immediate from Lynne Yengulalp. diff --git a/theorems/T000864.md b/theorems/T000864.md new file mode 100644 index 000000000..ac5444871 --- /dev/null +++ b/theorems/T000864.md @@ -0,0 +1,9 @@ +--- +uid: T000864 +if: + P000235: true +then: + P000236: true +--- + +Immediate from Lynne Yengulalp. diff --git a/theorems/T000865.md b/theorems/T000865.md new file mode 100644 index 000000000..a9e21ed60 --- /dev/null +++ b/theorems/T000865.md @@ -0,0 +1,9 @@ +--- +uid: T000865 +if: + P000063: true +then: + P000235: true +--- + +Immediate from Lynne Yengulalp.