Extreme sequence topology as coproduct

Question

Let $X$ be the set of sequences with only finitely many non-zero terms and metric $$ d(x,y)\triangleq \begin{cases} \sum_{n=1}^{\infty} |x_n-y_n| : & x_n \neq 0 \,\, \text{iff} \,\, y_n\neq 0\\ \infty :& \text{else}. \end{cases} $$

Is $X$ homeomorphic to $\coprod_{n \in \mathbb{N}} \mathbb{R}$?

Answer 1

The $d$ is not a "real" metric, as $\infty$ is a value, but it induces a valid topology. And $B_d(\underline{0},1)$ in $(X,d)$ is strongly infinite dimensional (homeomorphic to $\ell^1$, really), while the coproduct is just one-dimensional, so they're not homeomorphic.