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Proposition: Metric Spaces are Hausdorff Spaces
Let \((X,d)\) be a metric space. Then any two distinct points \(x,y\in X\) with \(x\neq y\) have also distinct neighborhoods \(U,V\), i.e. \(U\cap V=\emptyset\).
\((X,d)\) is then called a Hausdorff space.
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References
Bibliography
 Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984