Proof
(related to Proposition: Metric Spaces and Empty Sets are Clopen)
The arguments are as follows:
 \(X\) is open, because it is the neighborhood of any of its points.
 If \(X\) is open, then by definition, its complement \(X\setminus X=\emptyset\) is closed.
 It is vacuously true that the empty set \(\emptyset\) is open, because there is no \(x\in\emptyset\), for which \(\emptyset\) could (or could not) be the neighborhood.
 If \(\emptyset\) is open, then by definition, its complement \(X\setminus \emptyset=X\) is closed.
Therefore, \(X\) and \(\emptyset\) are clopen.
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References
Bibliography
 Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984