Traits for Cohen's modified product (S207)

properties/P000026.md

@@ -11,3 +11,8 @@ refs:
 A space with a countable dense subset.
 
 Defined on page 7 of {{zb:0386.54001}}.
+
+----
+#### Meta-properties
+
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.

properties/P000051.md

@@ -25,3 +25,6 @@ Defined as "dispersed" in e.g. {{zb:0164.53101}}.
 #### Meta-properties
 
 - This property is hereditary.
+- This property is preserved by finer topologies.
+- This property is preserved by arbitrary disjoint unions.
+- This property is preserved by finite products.

spaces/S000207/properties/P000051.md

@@ -0,0 +1,7 @@
+---
+space: S000207
+property: P000051
+value: true
+---
+
+$X$ is finer than $\omega_1 \times (\omega_1 + 1)$ with the product topology, {S35|P51} and {S36|P51}.

spaces/S000207/properties/P000062.md

@@ -0,0 +1,7 @@
+---
+space: S000207
+property: P000062
+value: false
+---
+
+Consider the open cover $\{[0,\alpha]\times (\omega_1 + 1) \mid \alpha \in \omega_1\}$. This has no dense countable subcollection by a similar argument given as in {S35|P62}.

spaces/S000207/properties/P000081.md

@@ -0,0 +1,7 @@
+---
+space: S000207
+property: P000081
+value: false
+---
+
+The subspace $\{0\}\times (\omega_1 + 1)\subseteq X$ is homeomorphic to {S155} and {S155|P81}.

spaces/S000207/properties/P000114.md

@@ -0,0 +1,7 @@
+---
+space: S000207
+property: P000114
+value: true
+---
+
+Since $|\omega_1|=|\omega_1 + 1| = \aleph_1$.