Locally orderable spaces are radial

properties/P000172.md

@@ -19,3 +19,7 @@ Compare with {P80}.
 
 - This property is hereditary.
 - $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
+- A space that is locally {P172} (every point has a neighborhood with the property)
+has the property.
+- This property is not preserved by products
+(Example: {S78|P172}).

theorems/T000774.md

@@ -0,0 +1,14 @@
+---
+uid: T000774
+if:
+  P000120: true
+then:
+  P000172: true
+---
+
+Let $A\subseteq X$ and $p\in\overline A$.
+Since $X$ is {P120}, $p$ has an open neighborhood $U$
+that is orderable ({P133}), and hence also {P172}
+[(Explore)](https://topology.pi-base.org/spaces?q=LOTS%2B%7ERadial).
+Because $p\in\overline{A\cap U}$, and thus $p\in\text{cl}_U(A\cap U)$,
+it follows that there is a transfinite sequence of points of $A\cap U$ converging to $p$.