= Solution
\b[False.] The real line with its usual <metric> is <complete> and is <homeomorphic> to $(0,1)$ with its usual <metric>, through
$$
h(x)=\frac12+\frac1\pi\arctan x.
$$
Its inverse is $h^{-1}(u)=\tan(\pi(u-1/2))$, so both maps are <continuous>. However, $(1/n)_{n\ge2}$ is a <Cauchy sequence> in $(0,1)$ and has no limit there: its only possible real limit is zero. Hence $(0,1)$ is not a <complete metric space>. \b[Completeness is a property of the <metric>, not just of its <topology>.]
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