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Closed Set

Basic definition

A closed set is a subset of a topological space that contains all of its limit points. A closed interval is an example of a closed set.

Detailed definition

There are several equivalent definitions of a closed set. Let S be a subset of a metric space. A set S is closed if
1. The complement of S is an open set, 2.S is its own set closure, 3. Sequences/nets/filters in S that converge do so within S, 4. Every point outside S has a neighborhood disjoint from S.

Related terms

Borel set | boundary point | Cantor set | closed ball | closed disk | closed interval | compact set | half-closed interval | open set

Educational grade level

college level

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