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The set of all limit points of a subset.
The derived set of a subset in a topological space is the set
where “limit point” is as in limit point.
The derived set records the accumulation behavior of , and it satisfies the basic relationship with closure:
A topological space is an ordered pair where is a set and is a topology on , meaning:
Here denotes the power set of , and the members of are the open sets (whose complements are the closed sets). Many standard constructions—such as the subspace topology, product topology, and quotient topology—produce new topological spaces from existing ones.
A limit point (or accumulation point) of a subset in a topological space is a point such that every neighborhood of satisfies
The set of all limit points of is the derived set of . Limit points also describe closure via the identity .
The closure of a subset in a topological space is