More traits with P243 (countable pi-weight)

Author: prabau

properties/P000243.md

@@ -21,5 +21,7 @@ Defined on page 14 of {{zb:0559.54003}}.
 
 - This property is hereditary with respect to dense sets.
 - This property is hereditary with respect to open sets.
+- If some open dense subset of $X$ has the property, then so does $X$.
 - $X$ satisfies this property iff its Kolmogorov quotient $\operatorname{Kol}(X)$ does.
 - This property is preserved by countable products.
+- This property is preserved by countable disjoint unions.

spaces/S000029/properties/P000243.md

@@ -0,0 +1,8 @@
+---
+space: S000029
+property: P000243
+value: true
+---
+
+{S27} is an open dense subspace of $X$
+and {S27|P243}.

spaces/S000031/properties/P000243.md

@@ -0,0 +1,8 @@
+---
+space: S000031
+property: P000243
+value: true
+---
+
+{S29|P243},
+and $X$ is a product of two copies of that space.

spaces/S000097/properties/P000243.md

@@ -0,0 +1,8 @@
+---
+space: S000097
+property: P000243
+value: true
+---
+
+{S131} is an open dense subspace of $X$
+and {S131|P243}.

spaces/S000139/properties/P000243.md

@@ -0,0 +1,10 @@
+---
+space: S000139
+property: P000243
+value: true
+---
+
+The set $X\setminus\{\infty\}$ is open and dense in $X$.
+It is {P243}
+because it is homemorphic to a countable disjoint union of intervals $(0,1)\cong\mathbb R$
+and {S25|P243}.

spaces/S000140/properties/P000243.md

@@ -0,0 +1,8 @@
+---
+space: S000140
+property: P000243
+value: true
+---
+
+{S25} is an open dense subspace of $X$
+and {S25|P243}.

spaces/S000156/properties/P000243.md

@@ -0,0 +1,8 @@
+---
+space: S000156
+property: P000243
+value: true
+---
+
+The set $\omega\times\omega$ is open dense in $X$ and discrete.
+So the singletons $\{x\}$ for $x\in\omega\times\omega$ form a countable $\pi$-base for $X$.

spaces/S000165/properties/P000243.md

@@ -0,0 +1,8 @@
+---
+space: S000165
+property: P000243
+value: true
+---
+
+{S23} is an open dense subspace of $X$
+and {S23|P243}.