Properties of $m\mathcal H$-compact sets in hereditary $m$-spaces

Let $(X, m, \mathcal{H})$ be a hereditary $m$-space. A subset $A$ of $X$ is said to be $\mathcal{H}$-compact relative to $X$ if for every cover $\mathcal U$ of $A$ by $m$-open sets of $X$, there exists a finite subset $\mathcal{U}_0$ of $\mathcal{U}$ such that $A \setminus \cup\ \mathcal{U}_0 \in$ $\mathcal{H}$. We obtain several properties of these sets. And also, we define and investigate two kinds of strong forms of $\mathcal{H}$-compact relative to $X$.