# Definition:Separation (Topology)

(Redirected from Definition:Partition (Topology))

Jump to navigation
Jump to search
## Definition

Let $T = \struct {S, \tau}$ 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
- $(2): \quad A \cup B = S$
- $(3): \quad A \cap B = \O$

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**.

- Definition:Separated Sets, a definition which is linked by Components of Separation are Separated Sets.

- Definition:Separable Space, an unrelated definition.

- Definition:Tychonoff Separation Axioms, a classification system for topological spaces.

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

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $6.2$: Connectedness: Definition $6.2.2$ - 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness