Work on $\omega_1$ (S35)

properties/P000087.md

@@ -17,3 +17,4 @@ Contrary to Munkres or Willard, we do not assume any separation axiom like {P3},
 #### Meta-properties
 
 - This property is preserved by arbitrary products.
+- This property is preserved by $\Sigma$-products.

properties/P000186.md

@@ -13,3 +13,4 @@ Homeomorphic to a subspace of a space that is {P87} and {P187}.
 
 - This property is hereditary.
 - This property is preserved by countable products.
+- This property is preserved by $\Sigma$-products.

spaces/S000035/properties/P000062.md

@@ -0,0 +1,10 @@
+---
+space: S000035
+property: P000062
+value: false
+---
+
+Consider the open cover $\mathcal{O}=\{[0,\alpha)\mid \alpha \in X\}$.
+Every countable subset of $\omega_1$ has an upper bound in $\omega_1$.
+So if $\mathcal U$ is a countable subcollection of $\mathcal O$, there is some $\alpha\in\omega_1$ such that
+$\bigcup\mathcal U\subseteq[0,\alpha]$, which is not dense in $X$.

spaces/S000035/properties/P000082.md

@@ -0,0 +1,7 @@
+---
+space: S000035
+property: P000082
+value: true
+---
+
+For $\alpha \in X$ the open neighborhood $[0,\alpha + 1)\subseteq X$ is {P57} and {P190} and thus {P53} [(Explore)](https://topology.pi-base.org/spaces?q=Countable+%2B+Ordinal+space+%2B+%7EMetrizable).