Metric space
Convergence

Convergence of sequences in Euclidean space is defined as follows:

Convergence of sequences in a topological space is defined as follows:

In metric spaces, both of these definitions make sense and they are equivalent. This is a general pattern for topological properties of metric spaces: while they can be defined in a purely topological way, there is often a way that uses the metric which is easier to state or more familiar from real analysis.
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