Let \(X\) be a topological space . It is called Hausdorff space if for arbitrary two distinct points \(p,q\in X\) there are two disjoint open subsets \(U\) and \(V\), such that \(p\in U\), \(q\in V\).
Remarks
- The idea is, that two points can be distinguished in a topological sense.
- Limits of sequences are unique in Hausdorff spaces.
- Finite subsets are closed in Hausdorff spaces.
Examples Link to heading
- Metric spaces are Hausdorff.
- Subspaces of Hausdorff spaces are Hausdorff.
- Product spaces of Hausdorff spaces are Hausdorff.
- Disjoint union of Hausdorff spaces is Hausdorff.
Warning
A quotient space
of a Hausdorff space is not Hausdorff in general. For example, the topology of \(\mathbb{R}/\mathbb{Q}\) is the trivial
one.