Abstract

For a Tychonoff space $X$, we consider the space $I_{f}(X)$ that is a subspace of the space of all idempotent probability measures on $X$. Using a cover $\gamma_{X}$ of the origin space $X$ we build an open cover $\gamma_{0}$ of $I_{f}(X)$. Then we show that if $\gamma_{X}$ is a finite-component family, then $\gamma_{0}$ is also a finite-component family. We obtain that the space $I_{f}(\beta\, X)$ of idempotent probability measures on the Stone-Čech compactification $\beta\,X$ of $X$ is a perfect compactification of $I_{f}(X)$. Finally, applying the superparacompactness criterion, we establish that $I_{f}(X)$ is superparacompact if and only if the given Tychonoff space $X$ is superparacompact.

Keywords: Superparacompact space; perfect compactification; idempotent probability measure; finite-component cover.

MSC: 28C10, 46G12, 54A05, 54D20

DOI: 10.57016/MV-CQnA1710

Pages:  1--9