The (infinite) symmetric product of a based topological space $(X,e)$, denoted by $\mathrm{SP}(X,e)$, 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 following is my question:

Provided that $X$ is metrizable, can we say that $\mathrm{SP}(X,e)$ is metrizable in general?

Cheers.