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

That is, $A$ 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.


 * 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.