Definition:Separation (Topology)

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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

  • Results about separations can be found here.


Sources