Metric topological groups: their metric approximation and metric ultraproducts

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We define a metric ultraproduct of topological groups with left-invariant metric, and show that there is a countable sequence of finite groups with left-invariant metric whose metric ultraproduct contains isometrically as a subgroup every separable topological group with left-invariant metric. In particular, there is a countable sequence of finite groups with left-invariant metric such that every finite subset of an arbitrary topological group with left-invariant metric may be approximated by all but finitely many of them. We compare our results with related concepts such as sofic groups, hyperlinear groups and weakly sofic groups.