Hausdorff Space

A Hausdorff space is a topological space in which any two distinct points can be enclosed in disjoint open neighborhoods. Also called the $T_2$ condition, this requirement is one of the separation axioms of topology. It expresses a distinction between points using open sets rather than numerical distances. Hausdorffness is especially important because it guarantees uniqueness of convergent limits and, together with compactness, supports several fundamental results about continuous maps. (people.math.osu.edu)

Definition and separation axioms

Let $X$ be a set with topology $\tau$. The space $(X,\tau)$ is Hausdorff if, whenever $x,y\in X$ and $x\ne y$, there exist open sets $U,V\in\tau$ such that

These sets are open neighborhoods of the respective points. Their disjointness is essential: merely finding a neighborhood of each point that excludes the other does not establish Hausdorffness. (people.math.osu.edu)

Every Hausdorff space is a $T_1$ space, meaning that each singleton is a closed set. To see this, fix $x$. For every $y\ne x$, separation supplies an open neighborhood of $y$ not containing $x$; their union is $X\setminus\{x\}$. The converse fails. Stronger conditions distinguish points from closed sets, or two closed sets from one another, giving the notions of regular spaces and normal spaces. Terminology varies: some authors include $T_1$ in these definitions, whereas others state it separately. (math.ucla.edu)

Examples and counterexamples

Every metric space is Hausdorff. If $d(x,y)=r>0$, the open balls of radius $r/3$ centered at $x$ and $y$ are disjoint by the triangle inequality. Thus Euclidean spaces, including the real line with its usual topology, satisfy the condition. (math.mit.edu)

An infinite set with the cofinite topology is $T_1$ but not Hausdorff. Every nonempty open set has finite complement, so any two nonempty open sets intersect. This illustrates why closed singletons are insufficient for disjoint-neighborhood separation. (math.mit.edu)

The line with two origins is another counterexample. Start with two copies of the real line and identify corresponding nonzero points, while keeping the origins distinct. Every neighborhood of either origin contains sufficiently small nonzero points, so neighborhoods of the two origins necessarily overlap. It also shows that a quotient of a Hausdorff space need not remain Hausdorff. (math.mit.edu)

Uniqueness of limits

In a Hausdorff space, a convergent sequence has exactly one limit. If a sequence converged to distinct points $x$ and $y$, disjoint neighborhoods $U$ and $V$ would force its sufficiently late terms to belong to both sets—an impossibility. (math.rice.edu)

The complete characterization uses a net, a generalization of a sequence indexed by a directed set:

If separation fails for $x\ne y$, one can select points in intersections of their neighborhoods, ordered by increasing refinement, to obtain a net converging to both. (math.rice.edu)

Uniqueness for sequences alone does not characterize Hausdorffness in arbitrary spaces. On an uncountable set with the countable-complement topology, convergent sequences have unique limits, although the space is not Hausdorff. For first-countable spaces, however, uniqueness of sequential limits does characterize the condition. (sam.nitk.ac.in)

The closed-diagonal characterization

Consider the Cartesian product $X\times X$, equipped with the product topology. Its diagonal is

Then $X$ is Hausdorff exactly when $\Delta_X$ is closed. Disjoint neighborhoods of distinct points produce an open rectangle avoiding the diagonal. Conversely, a basic open rectangle around $(x,y)$ that avoids the diagonal has disjoint coordinate neighborhoods. (math.mit.edu)

This characterization emphasizes that Hausdorffness is a global property of the topology: it describes how the equality relation sits inside the product space, without requiring a metric or a choice of coordinates. The product topology is crucial to the statement. (math.mit.edu)

Compactness and continuous maps

Every compact subset of a Hausdorff space is closed. For a point outside the subset, separate it from each point inside; compactness reduces the resulting cover to finitely many neighborhoods. Intersecting the corresponding neighborhoods of the outside point gives an open set avoiding the entire subset. (math.mit.edu)

Consequently, a continuous bijection from a compact space to a Hausdorff space is a homeomorphism. Closed subsets of the domain are compact, their images are compact and therefore closed, and this makes the inverse continuous. Compactness here means that every open cover has a finite subcover; Hausdorffness is an additional condition, not part of that definition. (math.mit.edu)

Behavior under constructions

Every subspace of a Hausdorff space is Hausdorff under the subspace topology. Arbitrary products of Hausdorff spaces are also Hausdorff: two distinct tuples differ in some coordinate, whose disjoint neighborhoods pull back to disjoint neighborhoods in the product. (math.ucla.edu)

By contrast, continuous images and quotient spaces need not preserve Hausdorffness, as the line with two origins demonstrates. Identifying points can prevent their equivalence classes from having disjoint open neighborhoods, even when the original space has that separation property. (math.mit.edu)