Definition:Separation (Topology)

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 the context of topology 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.

• Components of Separation are Clopen

• 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
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): separation (of a set)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): separation (of a set)