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Is this true?

Completeness is a topological property

No — it is false.

The counterexample

(0,1) and

Homeomorphic, but is complete and (0,1) is not (1/n is Cauchy).

The kind of mistake this is

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this
completenessmetric spaces

Others that fail the same way

From Metric Spaces › Completeness and Baire category

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