Right ray topologies on omega_1 are not injectively path connected

Author: prabau

properties/P000037.md

@@ -18,3 +18,4 @@ Compare with {P38} and {P95}.
 
 - $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is preserved by arbitrary products.
+- This property is preserved in any coarser topology.

properties/P000038.md

@@ -17,3 +17,8 @@ and {P95}, which requires the path to be a homeomorphism onto its image.
 Defined on page 29 of {{zb:0386.54001}} with the name "arc connected".
 Here we reserve that name for the stronger property {P95},
 which is the more common usage in the literature.
+
+#### Meta-properties
+
+- This property is preserved by arbitrary products.
+- This property is preserved in any coarser topology.

spaces/S000220/properties/P000038.md

@@ -0,0 +1,7 @@
+---
+space: S000220
+property: P000038
+value: false
+---
+
+{S221} has a coarser topology than $X$, but {S221|P38}.

spaces/S000221/properties/P000038.md

@@ -0,0 +1,12 @@
+---
+space: S000221
+property: P000038
+value: false
+---
+
+Let $f:[0,1]\to X$ be a path in $X$.
+Take a countable dense subset $E$ of $[0,1]$.
+Its image $f(E)$ is countable and thus bounded in $X$, say by some $u\in\omega_1$.
+So $f(E)$ is contained in the closed set $[0,u]$.
+And by density of $E$ in $[0,1]$, $f([0,1])$ is also contained in $[0,u]$.
+Since $[0,u]$ is countable, $f$ cannot be injective.