Hausdorff space - Revision history

InfernalAtom683 at 07:16, 11 April 2026

2026-04-11T07:16:30Z

InfernalAtom683 at 03:43, 6 April 2026

2026-04-06T03:43:50Z

InfernalAtom683: InfernalAtom683 moved page Hausdorffness to Hausdorff space

2026-03-24T01:55:32Z

InfernalAtom683 moved page Hausdorffness to Hausdorff space

InfernalAtom683: Created page with "A '''Hausdorff space''' (or ''' $T_2$ 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 $(X,\tau)$ is Hausdorff, if for any two points $x,y\in X$
2026-03-22T13:20:43Z

Created page with "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 sets containing each point. Consequently, Hausdorff property ensures that limits of sequences are unique when they exist. == Definitions == A topological space <math>(X,\tau)</math> is Hausdorff, if for any two points <math>x,y\in X</math..."

New page
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 ==
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|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|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|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 | 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 ==
Every metric space is a Hausdorff space.
{{Proof|proof=<div style="float:right; width:320px; margin:0 0 0.5em 1em;">
<kroki lang="tikz">
\documentclass[tikz,border=8pt]{standalone}
\usetikzlibrary{calc}
\usepackage{amsmath}
\begin{document}
\begin{tikzpicture}[scale=1.3]

\def\r{1.5}

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

\def\ang{20}

\coordinate (y) at ({2\rcos(\ang)}, {2\rsin(\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) -- ({\rcos(\ang)}, {\rsin(\ang)})
node[midway, above, sloped] {\footnotesize $r=\dfrac{d(x,y)}{2}$};

\end{tikzpicture}
\end{document}
</kroki></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.}}