Product of Unit Ball and Real Line Is Hoemomorphic to Vector Space

Question

Let $X$ be a normed vector space and let $B$ be the boundary of its unit ball. Let $d_X$ be the metric induced by the norm on $X$ restricted to $B$. Then, is the topology on $X\cong B\times \mathbb{R}$ with product metric $ d_X \times \|\cdot - \cdot\|? $

I was thinking, and I guess that the map $$ x \mapsto \begin{cases} (\frac{x}{\|x\|},\|x\|) & x \neq 0\\ 0 & x=0. \end{cases} $$ should define a homeomorphism from $X$ to $B\times \mathbb{R}$. However, beyond this point I'm confused since the latter looks like a cylender in my mind..but maybe I'm missing something...

Answer 1

This map is not even continuous. If $x_n \to 0$ in $X$ then $\frac {x_n} {\|x_n\|}$ need not even have a limit.

But your map is a homeomorphism from $X\setminus \{0\}$ onto $B \times (0,\infty)$.