Geometrical properties of the space of idempotent probability measures

Abstract

Although traditional and idempotent mathematics are "parallel'', by an application of the category theory we show that objects obtained the similar rules over traditional and idempotent mathematics must not be "parallel''. At first we establish for a compact metric space X the spaces P(X) of probability measures and I(X) idempotent probability measures are homeomorphic ("parallelism''). Then we construct an example which shows that the constructions P and I form distinguished functors from each other ("parallelism'' negation). Further for a compact Hausdorff space X we establish that the hereditary normality of I<sub>3</sub>(X)\ X implies the metrizability of X.