The (infinite) symmetric product of a based topological space $(X,e)$, denoted by $SP(X,e)$, can be viewed as the topological space of ''multisets'' in $X$ containing the base point $e$ infinitely many times (please see http://en.wikipedia.org/wiki/Infinite_symmetric_product for the precise definition). The questions I would like to ask you guys are the following:
(A) If $X$ is metrizable, can we say that $SP(X,e)$ is metrizable in general?
(B) $SP(X,e)$ can naturally be viewed as a monoid (the monoid operation being the ''sum'' or ''union'' of multisets), but is it a topological monoid in general? That is, is the monoid operation continuous in general?
Cheers.
