Geometrical properties of space $I_f(X)$ of idempotent probability measures

TL;DR

This paper investigates the geometrical properties of the space of idempotent probability measures, establishing conditions under which certain topological properties are preserved by the functor $I_f$.

Contribution

It proves that the functor $I_f$ preserves ANR properties and certain manifold structures like $Q$-manyfolds and Hilbert cubes.

Findings

$I_f(X)$ is an ANR if and only if $X$ is an ANR.

$I_f$ preserves properties of being a $Q$-manyfold or Hilbert cube.

$I_f$ preserves properties of map layers related to ANR-compactness.

Abstract

In the paper we proved that for a compact $X$ inclusion $I_{f}(X)\in ANR$ holds if and only if $X\in ANR$. Further, it is shown that the functor $I_{f}$ preserves property of a compact to be $Q$-manyfold or a Hilbert cube, preserves property of map layers to be $ANR$-compact, compact $Q$-manyfold and a Hilbert cube (the finite sum of Hilbert cubes)