(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.
- Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984