Are there any examples of COUNTABLE topological groups with a non-discrete topology which are metrizable and complete? I think such do not exist.
2026-03-25 01:31:25.1774402285
Countable complete topological group
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No. Any nonempty countable complete metric space has an isolated point (if it didn't, then each singleton would have empty interior but the union of countably many singletons is the whole space, violating the Baire category theorem). If your space is a topological group, this then implies it is discrete.