If $X$ is T1 with a non-empty finite open subset, does $X$ have isolated points?

general-topology

If $x \in V$ then $F:=V\setminus \{x\}$ is finite (as a subset of $V$) and so closed (as $X$ is $T_1$). So $\{x\} = V\cap (X\setminus F)$ is open in $X$ and so $x$ is isolated.

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