= Solution
{c}
\b[$X$ is not <compact>.] In the <subspace topology> on $X=[0,1]$, consider the <open cover>
$$
\mathcal U=\{[0,1-1/n):n\geq2\}\cup\{\{1\}\}.
$$
The <intervals> are basic <open sets> in the ambient <Sorgenfrey line>, and $\{1\}=X\cap[1,2)$ is <open> in $X$. Every $x<1$ belongs to one of those <intervals>, while the <singleton> covers $1$.
A <finite subcover> would have to include $\{1\}$. Among its finitely many other members, let $N$ be the largest index. Their <union> is $[0,1-1/N)$, leaving $1-1/N$ uncovered. Selecting no such member also fails to cover $0$. Hence no <finite subcover> exists.
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