Hausdorff space: Difference between revisions
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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|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>. | 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|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 | 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 == | == Examples == | ||
Revision as of 15:16, 11 April 2026
A Hausdorff space (or 
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 sets containing each point. Consequently, Hausdorff property ensures that limits of sequences are unique when they exist.
Definitions
A topological space 
is Hausdorff, if for any two points 
, there exists two disjoint open sets 
, 
, such that 
and 
.
Equivalent Definitions
Any convergent sequence in 
has at most one limit.
Properties
Subspaces of Hausdorff spaces are Hausdorff.
Let 
with Hausdorff. For 
, there exist disjoint open sets 
with 
and 
. Then 
and 
are disjoint open sets in 
containing 
and 
.
Finite products of Hausdorff spaces are Hausdorff.
Let 
be Hausdorff. Consider points 
. There exists an index 
with 
.
Since 
is Hausdorff, choose disjoint open sets 
containing 
and 
. Then 
and 
are disjoint open sets containing the two points.
Compact subsets of Hausdorff spaces are closed.
Let 
be compact and Hausdorff. For any 
, for each 
choose disjoint open sets 
and 
.
The collection 
covers 
. By compactness, finitely many 
cover . Then
, and is disjoint from . Hence is closed.
Examples
Every metric space is a Hausdorff space.
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Diagram-Code:
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Let 
be a metric space, take two distinct points 
such that 
.
Consider open balls 
and 
where 
. The open balls are both open with 
and 
.
For contradiction, assume there exists 
, then 
and 
. By triangular inequality,
Thus, 
; therefore is Hausdorff.
See also
Terminology
| en | fr | de | zh | ja | |
|---|---|---|---|---|---|
| Hausdorff space | espace de Hausdorff (espace séparé) | hausdorff-Raum (hausdorffscher Raum) | Hausdorff 空间 | Hausdorff 空間 | ハウスドルフ空間 |