Definition:Hausdorff Space

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Definition

Let $T = \struct {S, \tau}$ be a topological space.


Definition 1

$\left({S, \tau}\right)$ is a Hausdorff space or $T_2$ space if and only if:

$\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U \cap V = \varnothing$

That is:

for any two distinct elements $x, y \in S$ there exist disjoint open sets $U, V \in \tau$ containing $x$ and $y$ respectively.


Definition 2

$\left({S, \tau}\right)$ is a Hausdorff space or $T_2$ space if and only if each point is the intersection of all its closed neighborhoods.


Definition 3

$\left({S, \tau}\right)$ is a Hausdorff space or $T_2$ space if and only if:

$\forall x, y \in S, x \ne y: \exists N_x, N_y \subseteq S: \exists U, V \in \tau: x \subseteq U \subseteq N_x, y \subseteq V \subseteq N_y: N_x \cap N_y = \varnothing$

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.


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.


Equivalence of Definitions

See Equivalence of Definitions of $T_2$ Space for a proof that these definitions are equivalent.

Note that while some sources give Definition 1 and others Definition 3, it is rarely indicated specifically that the definitions are equivalent.

In particular, there may exist places on the internet where a page using Definition 3 may directly link to another page which uses Definition 1 without comment that the definitions are linguistically different.


Also see

  • Results about $T_2$ (Hausdorff) spaces can be found here.


Source of Name

This entry was named for Felix Hausdorff.


Sources