# Metric Space is Hausdorff

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## Theorem

Let $M = \struct {A, d}$ be a metric space.

Then $M$ is a Hausdorff space.

## Proof 1

Let $x, y \in A: x \ne y$.

Then from Distinct Points in Metric Space have Disjoint Open Balls, there exist open $\epsilon$-balls $\map {B_\epsilon} x$ and $\map {B_\epsilon} y$ which are disjoint open sets containing $x$ and $y$ respectively.

Hence the result by the definition of Hausdorff space.

$\blacksquare$

## Proof 2

This article, or a section of it, needs explaining.Do we really need this second proof? It's the same argument as Distinct Points in Metric Space have Disjoint Open Balls/Proof 2 and Open Balls whose Distance between Centers is Twice Radius are Disjoint.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

Aiming for a contradiction, suppose $M$ is not Hausdorff.

That is, there are $x, y \in A: x \ne y$ such that:

- $\forall \epsilon \in \R_{>0}: \exists z \in \map {B_\epsilon} x \cap \map {B_\epsilon} y$

where $\map {B_\epsilon} x$ denote the open $\epsilon$-ball of $x$ in $M$.

Let $r = \dfrac {\map d {x, y} } 2$.

Let $z \in \map {B_r} x \cap \map {B_r} y$.

Then:

\(\ds z\) | \(\in\) | \(\ds \map {B_r} x \cap \map {B_r} y\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds z\) | \(\in\) | \(\ds \map {B_r} x\) | Definition of Set Intersection | ||||||||||

\(\, \ds \land \, \) | \(\ds z\) | \(\in\) | \(\ds \map {B_r} y\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds \map d {x, z}\) | \(<\) | \(\ds r\) | Definition of Open Ball | ||||||||||

\(\, \ds \land \, \) | \(\ds \map d {y, z}\) | \(<\) | \(\ds r\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds \map d {x, z} + \map d {y, z}\) | \(<\) | \(\ds 2 r\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds \map d {x, z} + \map d {y, z}\) | \(<\) | \(\ds \map d {x, y}\) |

This contradicts Metric Space Axiom $\text M 2$.

Thus, $M$ has to be Hausdorff.

$\blacksquare$

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $4$: The Hausdorff condition: $4.2$: Separation axioms: Example $4.2.3$ - 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces