The set of all cluster or limit points of a set
is called the derived set ofand is labelled
More precisely, given a set
the cluster set can be defined as
for any neighbourhood
of
Example: Consider the set
The only cluster point ofis
so the derived set ofis
Example: If we have the indiscrete topology
on a set
For any element
is the only open set containingWe have
Example: Suppose we have the discrete topology for which all subsets ofare open sets. For any elementis the only open set containingNo element of the discrete set is a limit point since
is a neighberhood of
but contains no elements ofapart fromHence