Suppose $X$ is a topological space satisfying the following property: every (nonempty) open set of $X$ has a (nonempty) proper open subset.
Does this property have a name?
What are some spaces with this property?
What are some properties that imply this property (eg. Hausdorff, separable, second countable, etc.)?
The intuition is that, if you have an open set $U \subseteq X$, you can "zoom in" at any point of $U$, forever.
Example. If $X$ has the discrete topology, then it doesn't have this property, because singletons $\{x\} \subseteq X$ are open sets that don't have (nonempty) proper open subsets.
Example. If $X$ is ${\mathbb R}^n$ with the Euclidean topology, then it does have this property, because, for any open ball $B$ of radius $\epsilon$ around $x \in X$, you can take the open ball of radius $\epsilon/2$, which is a (nonempty) proper open subset of $B$.