On $Z$-Sets in the Space of Idempotent Probability Measures

It is shown that the spaces of probability measures and of idempotent probability measures on the same metrizable compact space are homeomorphic. An example is given which shows that the functor of probability measures differs from that of idempotent probability measures. A criterion for the metrizability of a compact Hausdorff space in terms of the functor of idempotent probability measures is obtained. Subspaces of spaces of idempotent probability measures that are $Z$-sets are distinguished.