# Definition:Separation (Topology)

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## Contents

## Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $A$ and $B$ be open sets of $T$.

$A$ and $B$ form a **separation of $T$** if and only if:

- $(1): \quad A$ and $B$ are non-empty
- $(2): \quad A \cup B = S$
- $(3): \quad A \cap B = \varnothing$

That is, if and only if $A$ and $B$ form a partition of the set $S$.

Such a **separation** can be denoted:

- $A \mid B$

$A$ and $B$ are said to **separate $T$**.

## Also known as

A **separation** in this particular context is also known as a **partition**.

However, because the latter term has a definition in set theory, **separation** is preferred so as to reduce ambiguity and the possibility of confusion.

## Also see

- Definition:Connected Topological Space, one of whose properties is
**admitting no separation**.

- Results about
**separations**can be found here.

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 4$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $6.2$: Connectedness: Definition $6.2.2$