Let $S$ be a subset of some topological space $X$.
What is known about the following property of $S$? Does it have a name? Is it standard?
Property: For all $x\in \bar{S}$ (the closure of $S$), and for all open sets $U$ with $x\in U$, there exists an open set $V$ with $x\in V\subseteq U$ such that $V\cap S$ is connected.
If $S$ is closed, then this just means that $S$ is locally connected. But for non closed $S$, the two concepts do not agree. (Both $\mathbb{R}\setminus \{0\}$ and its closure in $\mathbb{R}$ are locally connected, but $\mathbb{R}\setminus \{0\}$ does not possess this property in $\mathbb{R}$.)