For the functor of upper semicontinuous capacities in the category of compact Hausdorff spaces and two of its subfunctors, we prove open mapping theorems. These are counterparts of the open mapping theorem for the probability measure functor proved by Ditor and Eifler.

Let $𝔐$ and $\Re$ be algebras of subsets of a set $\Omega$ with $𝔐\subset \Re$, and denote by $E\left(\mu \right)$ the set of all quasi-measure extensions of a given quasi-measure $\mu$ on $𝔐$ to $\Re$. We give some criteria for order boundedness of $E\left(\mu \right)$ in $ba\left(\Re \right)$, in the general case as well as for atomic $\mu$. Order boundedness implies weak compactness of $E\left(\mu \right)$. We show that the converse implication holds under some assumptions on $𝔐$, $\Re$ and $\mu$ or $\mu$ alone, but not in general.

Let $𝔐$ and $\Re$ be algebras of subsets of a set $\Omega$ with $𝔐\subset \Re$, and denote by $E\left(\mu \right)$ the set of all quasi-measure extensions of a given quasi-measure $\mu$ on $𝔐$ to $\Re$. We show that $E\left(\mu \right)$ is order bounded if and only if it is contained in a principal ideal in $ba\left(\Re \right)$ if and only if it is weakly compact and $extrE\left(\mu \right)$ is contained in a principal ideal in $ba\left(\Re \right)$. We also establish some criteria for the coincidence of the ideals, in $ba\left(\Re \right)$, generated by $E\left(\mu \right)$ and $extrE\left(\mu \right)$.