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In a metric space, a point lies in the closure of a set iff it is the limit of a sequence from the set.
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.
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.
The closure of a subset in a topological space is
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.