= Solution
{c}
Let $U$ be <open> in the <Euclidean topology> on the <real line>, and let $x\in U$. There is an $\varepsilon>0$ with $(x-\varepsilon,x+\varepsilon)\subseteq U$. The basic <open set> $[x,x+\varepsilon)$ contains $x$ and is contained in $U$. Taking the <union> of these basic <neighbourhoods> over $x\in U$ shows that $U\in\mathcal T$. Thus
$$
\boxed{\mathcal T_{\mathrm{usual}}\subseteq\mathcal T.}
$$
In fact this is a strictly <finer topology>: $[0,1)$ is <open> in $\mathcal T$ but not in the <Euclidean topology>, since no usual <open ball> about $0$ is contained in it.
Back to article page
