## The Derived Set

The set of all cluster or limit points of a setis called the derived set ofand is labelled More precisely, given a setthe cluster set can be defined asfor any neighbourhoodof

Example: Consider the setThe only cluster point ofisso the derived set ofis

Example: If we have the indiscrete topologyon a setFor any elementis 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 sinceis a neighberhood ofbut contains no elements ofapart fromHence