Homotopy properties of the space $O{S}_f(X)$

Abstract

For a given compact Hausdorff space $X$, we construct the space $O{S}_f(X)$ of normed, order-preserving, weakly additive, positively homogeneous and semi-additive functionals (for brevity, semi-additive functionals) and it is proved that the hyperspace $\exp X$ of the space $X$ is a deformation retract of the constructed space. Further we show that the shapes of the spaces $O{S}_f(X)$ and $\exp X$ coincide. We establish that if the space $\exp X$ is contractible, then the space $O{S}_f(X)$ is also contractible.

12 pages