I came across the notion of a $\textit{topology induced by a norm}$.
If $(X,\Vert\ . \Vert)$ is a normed space w.r.t a norm $\Vert\ . \Vert: X \to \mathbb{R}$. Most sources define the topology $\tau$ on $X$ induced by $\Vert\ . \Vert$ as the sets $U \subset X$ open w.r.t. the metric $d: X \times X \to \mathbb{R}$ given by $d(x,y) = \Vert x – y \Vert$.
But would I be correct in assuming that an equivalent definition would be $\tau = \{\Vert\ . \Vert^{-1}(U) \mid U \subset \mathbb{R}\ \textrm{open} \}$?
Best Answer
With your version and $\Bbb R$ with norm $|\cdot|$, the set $(-3,-2)\cup(2,3)$ would be open, but $(2,3)$ not.