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- Jun 20, 2014

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$$X\setminus G = \bigcup_{y\neq x} X\setminus B_{n(y)} \supset \bigcup_{y\neq x} V_y \supset X\setminus\{x\}$$

then $\{x\} = G$. Hence, $\{x\}$ is a $G_\delta$ set.