= Solution
{c}
The two conditions for a <basis of a topology> both hold. First, every <real number> $x$ belongs to $[x,x+1)$, so the proposed basic <intervals> cover the <real line>. Second, if
$$
x\in[a,b)\cap[c,d),
$$
then $\max(a,c)\leq x<\min(b,d)$, and
$$
[a,b)\cap[c,d)=[\max(a,c),\min(b,d))
$$
is itself one of the proposed basic <intervals>. Thus every point in an <intersection> of two basic <sets> has a basic <neighbourhood> contained in that <intersection>.
Consequently the arbitrary <unions> of members of $\mathcal B$, including the <empty union>, form a <topology> $\mathcal T$. The <basis of a topology> conditions give closure under finite <intersections>; arbitrary <unions> are built into the definition. \b[This is the <lower limit topology>, also called the <Sorgenfrey line>.]
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