Derived set - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

The derived set of a subset $A\subseteq X$ in a topological space is the set

A'=\{x\in X : x \text{ is a limit point of } A\},

where “limit point” is as in limit point.

Relation to closure

The derived set records the accumulation behavior of $A$ , and it satisfies the basic relationship with closure:

\overline{A}=A\cup A'.

Examples