Local separating point

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Let $(X,\tau)$ be a locally compact separable metric space. We say a point $p \in X$ is a local separating point of $X$ provided there exists an open set $U \in \tau$ with $p \in U$ and two points $x$ and $y$ of the component of $U$ containing $p$ such that $U \setminus \{p\}$ is the sum of two mutually separated sets, one containing $x$ and one containing $y$.