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In a metric space, a set is closed iff it contains limits of all convergent sequences from it.
Sequential characterization of closed sets: Let be a metric space and let . Then is closed if and only if for every convergent sequence in with in , one has .
This is the standard “sequentially closed equals closed” principle for metric spaces, and it pairs naturally with the sequential characterization of closure.
A metric space is a pair where is a set and is a metric on .
Every metric space determines a topological space via the metric-induced topology, whose basic neighborhoods are open balls.
A closed set in a topological space is a subset such that its complement (the set difference) is a open set.
Closed sets are the natural targets of the closure operation: the closure of is the smallest closed set containing . Closed sets also provide an equivalent formulation of continuity via preimages of closed sets.
A convergent sequence in a topological space converges to a point if for every neighborhood of , there exists such that for all .
In a metric space , this is equivalent to . In a Hausdorff space, limits of convergent sequences are unique (see uniqueness of limits).
Sequential characterization of closure: Let be a metric space, let , and let . Then
where denotes the closure of .
This turns the topological notion of closure into a sequential condition in metric spaces, and it complements the sequential characterization of closed sets.