I want to ask a follow up question to the one I asked earlier here.
In Robert Israel's answer, it's posited that the natural embedding $\iota$ of $B = \{y \in \ell_\infty: \|y\| \le 1\}$ into $P = \prod_{x \in \ell_1} [-\|x\|,\|x\|]$ is a homeomorphism. Coming back to it a few days later, I realize I don't quite see immediately why this is. Is there a more explicit reason that such a map is a homeomorphism?