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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 closed set in a topological space is a subset whose complement is a open set.
A sequence in a topological space converges to if, for every neighborhood of , there is such that for all .
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.