View source for Hausdorff space

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A '''Hausdorff space''' (or '''<math>T_2</math> space''') is a type of [[topological space]] in which points can be "cleanly separated" by neighborhoods. Specifically, for any two distinct points, there exist disjoint [[Open set|open sets]] containing each point. Consequently, Hausdorff property ensures that limits of sequences are unique when they exist.

== Definitions ==
{{Definition|
A topological space <math>(X,\tau)</math> is Hausdorff, if for any two points <math>x,y\in X</math>, there exists two disjoint open sets <math>U,V\in \tau</math>, <math>U\cap V=\varnothing</math>, such that <math>x\in U</math> and <math>y\in V</math>.
}}
=== Equivalent Definitions ===
Any convergent [[sequence]] in <math>X</math> has at most one limit.

== Properties ==
{{Property
 |property=Subspaces of Hausdorff spaces are Hausdorff.
 |proof=Let <math>Y \subseteq X</math> with <math>X</math> Hausdorff. For <math>y_1, y_2 \in Y, y_1 \neq y_2</math>, there exist disjoint open sets <math>U, V \subseteq X</math> with <math>y_1 \in U</math> and <math>y_2 \in V</math>. Then <math>U \cap Y</math> and <math>V \cap Y</math> are disjoint open sets in <math>Y</math> containing <math>y_1</math> and <math>y_2</math>.
}}

{{Property
 |property=Finite products of Hausdorff spaces are Hausdorff.
 |proof=Let <math>X_1, \dots, X_n</math> be Hausdorff. Consider points <math>(x_1, \dots, x_n) \neq (y_1, \dots, y_n)</math>. There exists an index <math>i</math> with <math>x_i \neq y_i</math>.

Since <math>X_i</math> is Hausdorff, choose disjoint open sets <math>U_i, V_i \subseteq X_i</math> containing <math>x_i</math> and <math>y_i</math>. Then <math>U_1 \times \dots \times U_n</math> and <math>V_1 \times \dots \times V_n</math> are disjoint open sets containing the two points.}}

{{Property
 |property=Compact subsets of Hausdorff spaces are closed.
 |proof=Let <math>K \subseteq X</math> be compact and <math>X</math> Hausdorff. For any <math>x \in X \setminus K</math>, for each <math>y \in K</math> choose disjoint open sets <math>U_y\ni x</math> and <math>V_y \ni y</math>. 

The collection <math>\{V_y \mid y \in K\}</math> covers <math>K</math>. By compactness, finitely many <math>V_{y_1}, \dots, V_{y_n}</math> cover <math>K</math>. Then <math display="block">U = \bigcap_{i=1}^n U_{y_i}</math> is open, contains <math>x</math>, and is disjoint from <math>K</math>. Hence <math>K</math> is closed.
}}

== Examples ==
{{Example
 |example=Every [[metric space]] is a Hausdorff space.
 |proof=<div style="float:right; width:320px; margin:0 0 0.5em 1em;">
<tikz>
\def\r{1.5}

\coordinate (x) at (0,0);

\def\ang{20}

\coordinate (y) at ({2*\r*cos(\ang)}, {2*\r*sin(\ang)});

\draw[fill=blue!25!cyan, opacity=0.2, dashed, thick] (x) circle (\r);
\draw[fill=blue!25!cyan, opacity=0.2, dashed, thick] (y) circle (\r);

\fill (x) circle (2pt);
\fill (y) circle (2pt);

\node[below left] at (x) {$x$};
\node[above right] at (y) {$y$};

\draw[black, dashed, thick]
(x) -- ({\r*cos(\ang)}, {\r*sin(\ang)})
node[midway, above, sloped] {\footnotesize $r=\dfrac{d(x,y)}{2}$};

</tikz>
</div>
Let <math>(X,d)</math> be a metric space, take two distinct points <math>x,y</math> such that <math>d(x,y)>0</math>.

Consider open balls <math>B(x,r)</math> and <math>B(y,r)</math>where <math>r=\frac{d(x,y)}{2}>0</math>. The open balls are both open with <math>x\in B(x,r)</math> and <math>y\in B(y,r)</math>.

For contradiction, assume there exists <math>z\in B(x,r)\cap B(y,r)</math>, then <math>d(x,z)<r</math> and <math>d(y,z)<r</math>. By triangular inequality, <math display="block">d(x,y)\leq d(x,y)+d(x,z)<r+r=d(x,y),</math> there exists a contradiction.

Thus, <math>B(x,r)\cap B(y,r)=\varnothing</math>; therefore <math>(X,d)</math> is Hausdorff.
}}

== See also ==

* [[Topological space]]
* [[Convergence]]

{{Terminology
|rows=
{{Terminology/row
 |english=Hausdorff space
 |french=espace de Hausdorff
 |german=hausdorff-Raum
 |japanese=ハウスドルフ空間
 |chinese=Hausdorff 空间 · Hausdorff 空間
 |highlight=yes
}}
}}