Hausdorff space

• categories: Real Analysis, Definition

In topology, a Hausdorff space is a type of topological space that satisfies a key separation property. Specifically, any two distinct points in the space can be separated by disjoint open sets.

Formal Statement

A topological space $X$ is Hausdorff if for every $x,y\in X$ with $x\ne y$, there exist open sets $U,V\subseteq X$ such that:

$$x\in U,y\in V,\text{and}U\cap V=\varnothing$$

Intuition

The Hausdorff condition ensures that points in the space are “separated” in a strong sense, allowing us to distinguish between them with open sets. This is a fundamental property in many areas of topology and analysis because it guarantees certain desirable behaviors, such as the uniqueness of limits

Examples

1. Euclidean space ${\mathbb{R}}^n$: Every Euclidean space is a Hausdorff space. For any two distinct points in ${\mathbb{R}}^n$, we can always find two disjoint open balls around them, satisfying the Hausdorff condition.

Properties

• Uniqueness of Limits: One of the key consequences of the Hausdorff property is that limits of sequences (or nets) in a Hausdorff space are unique. If a sequence converges to a point, it can converge to only one point in the space.

• Compactness: In a Hausdorff space, any compact subset is closed. This is an important result that connects two fundamental topological concepts: compactness and separation.

• Metric space: Every metric space (where the distance between points is defined) is Hausdorff. The disjoint open sets in this case are typically open balls centered at each point.