Let \(X\) be a Hausdorff space . Then finite subsets of \(X\) are closed .
Proof
Consider a singelton \(\{p\}\) and show \(X\setminus \{p\}\) is open. Using the Hausdorff property we find for every point in \(X\setminus \{p\}\) a neighbourhood which is contained in \(X\setminus \{p\}\).