Property S with subsets
Theorem
Let $(X,d)$ be a metric space and let $Y \subset X$ have property S. Then for any $Z$ such that $Y \subset Z \subset \overline{Y}$, $Z$ has property $S$ and also $Z$ is a Peano space.
Let $(X,d)$ be a metric space and let $Y \subset X$ have property S. Then for any $Z$ such that $Y \subset Z \subset \overline{Y}$, $Z$ has property $S$ and also $Z$ is a Peano space.
