Definition:Hausdorff Space/Definition 3
Definition
Let $T = \struct {S, \tau}$ be a topological space.
$\struct {S, \tau}$ is a Hausdorff space or $T_2$ space if and only if:
- $\forall x, y \in S, x \ne y: \exists N_x, N_y \subseteq S: \exists U, V \in \tau: x \subseteq U \subseteq N_x, y \subseteq V \subseteq N_y: N_x \cap N_y = \O$
That is:
- for any two distinct elements $x, y \in S$ there exist disjoint neighborhoods $N_x, N_y \subseteq S$ containing $x$ and $y$ respectively.
That is:
- $\struct {S, \tau}$ is a $T_2$ space if and only if every two elements in $S$ are separated by neighborhoods.
Also known as
This condition is known as the Hausdorff condition.
For short, $T$ is Hausdorff is used to mean $T$ is a Hausdorff space.
Conveniently, a topological space is Hausdorff if any two distinct points can be housed off from one another in separate disjoint open sets.
Some sources use the term separated space for Hausdorff space but this is discouraged as there already exists considerable confusion and ambiguity around the definition of the word separated in the context of topology.
Some authors require a space to be Hausdorff before allowing it to be classed as a topological space, but this approach is unnecessarily limiting.
Also note that a Hausdorff space is the same thing as $T_2$ space.
Also see
Source of Name
This entry was named for Felix Hausdorff.
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $3$: Topological Spaces: $\S 3$: Neighborhoods and Neighborhood Spaces: Definition $3.3$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Hausdorff space
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Hausdorff space