= Solution
{c}
The usual <real line> is <connected>. In contrast,
$$
\mathbb R_{\mathcal T}=(-\infty,0)\sqcup[0,\infty)
$$
is a <separation of a topological space>. Both pieces are nonempty and disjoint; the first is <open> by part (c), and the second is the <open> <union> $\bigcup_{n\geq1}[0,n)$. Thus the <Sorgenfrey line> is not <connected>. Since a <homeomorphism> preserves <connectedness>, \b[the two <topological spaces> are not <homeomorphic>.] Merely showing that their <topologies> differ would not by itself exclude some other <homeomorphism> between them.
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