= Solution
{c}
The answers are
$$
\boxed{\text{yes, no, no, no}.}
$$
Distinct <natural numbers> have $d(x,y)>1$. Taking $\varepsilon=1$ in the <Cauchy sequence> condition forces every sufficiently late pair of terms to be equal. Every <Cauchy sequence> is therefore eventually constant and converges in $\mathbb N$, proving <completeness>.
The <sequence> $x_n=n$ has no <convergent subsequence>, since convergence would force its sufficiently late terms to lie within distance $1$ of its <limit of a sequence>, hence to equal that limit. Thus <sequential compactness> fails.
Every <singleton> is an <open ball> of radius $1$, so the <metric topology> is the <discrete topology>. For example, $\{1\}$ and $\mathbb N\setminus\{1\}$ form a separation into two nonempty <open sets>, so $X$ is not <connected>. It is consequently not <path-connected> either.
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