Add three new theorems regarding countable pi-character

properties/P000243.md

@@ -20,3 +20,5 @@ Defined on page 14 of {{zb:0559.54003}}.
 #### Meta-properties
 
 - This property is hereditary with respect to dense sets.
+- $X$ satisfies this property iff its Kolmogorov quotient $\operatorname{Kol}(X)$ does.
+- This property is preserved by countable products.

properties/P000244.md

@@ -23,3 +23,5 @@ Defined on page 16 of {{zb:0559.54003}}.
 #### Meta-properties
 
 - This property is hereditary with respect to dense sets.
+- $X$ satisfies this property iff its Kolmogorov quotient $\operatorname{Kol}(X)$ does.
+- This property is preserved by countable products.

spaces/S000108/properties/P000243.md

@@ -0,0 +1,7 @@
+---
+space: S000108
+property: P000243
+value: true
+---
+
+The set $\{\{n\}: n < \omega\}$ is a countable $\pi$-base for $\beta \omega$.

spaces/S000111/properties/P000243.md

@@ -0,0 +1,7 @@
+---
+space: S000111
+property: P000243
+value: true
+---
+
+$X$ is a dense subspace of {S108} and {S108|P243}.

spaces/S000216/properties/P000243.md

@@ -0,0 +1,7 @@
+---
+space: S000216
+property: P000243
+value: true
+---
+
+$X$ is a dense subspace of {S108} and {S108|P243}.

theorems/T000902.md

@@ -1,9 +1,20 @@
 ---
 uid: T000902
 if:
-  P000027: true
+  and:
+  - P000244: true
+  - P000026: true
 then:
   P000243: true
+refs:
+  - zb: "0559.54003"
+    name: Cardinal functions I (R. Hodel), Ch. 1 of Handbook of set-theoretic topology
 ---
 
-A base for the topology is a $\pi$-base.
+Let $A$ be a dense subset of $X$.
+For each $x\in A$, let $\mathcal V_x$ be a countable local $\pi$-base for $x$.
+Then $\bigcup\{\mathcal V_x:x\in A\}$ is a countable (global) $\pi$-base.
+
+To see this, if $O$ is a nonempty open set, there is some $x\in A\cap O$. Hence $O$ contains some $V\in\mathcal V_x$.
+
+This is a special case of Theorem 3.8(b) of {{zb:0559.54003}}.

theorems/T000903.md

@@ -0,0 +1,16 @@
+---
+uid: T000903
+if:
+  and:
+  - P000005: true
+  - P000029: true
+  - P000191: true
+  - P000244: true
+then:
+  P000163: true
+refs:
+  - zb: "0559.54003"
+    name: Cardinal functions I (R. Hodel), Ch. 1 of Handbook of set-theoretic topology
+---
+
+This is a special case of Corollary 6.4 of {{zb:0559.54003}}.

theorems/T000904.md

@@ -0,0 +1,15 @@
+---
+uid: T000904
+if:
+  and:
+  - P000087: true
+  - P000244: true
+then:
+  P000028: true
+---
+
+Because {T347}, it suffices to check the identity element $e$ has a countable local basis. Let $\mathcal{U}$ be a countable local $\pi$-base around $e$. Put $\mathcal{W} = \{U \cdot U^{-1}: U \in \mathcal{U}\}.
+
+We claim $\mathcal{W}$ is a countable local basis around $e$. It is clear that every element of $\mathcal{W}$ is a nbhd of $e$. If $O$ is an open nbhd of $e$, one can find an open nbhd $W$ of $e$ so that $W^2 \subseteq O$ by continuity of $(x, y) \mapsto x \cdot y$, and then $V = W \cap W^{-1}$ is open with $e \in V$.
+
+By hypothesis, there is $U \in \mathcal{U}$ so that $U \subseteq V$. Then one can easily verify that $U \cdot U^{-1} \subseteq O$.

theorems/T000905.md

@@ -0,0 +1,15 @@
+---
+uid: T000905
+if:
+  and:
+  - P000134: true
+  - P000016: true
+  - P000081: true
+then:
+  P000244: true
+refs:
+  - zb: "0559.54003"
+    name: Cardinal functions I (R. Hodel), Ch. 1 of Handbook of set-theoretic topology
+---
+
+This is a special case of Theorem 7.13 of {{zb:0559.54003}}.