Communications of the
Korean Mathematical Society CKMS

Let $(X, m$, $\mathcal{H})$ be a hereditary $m$-space and $\gamma : m \rightarrow P(X)$ be an operation on $m$. A subset $A$ of $X$ is said to be $\gamma\mathcal{H}$-compact relative to $X$ \cite{Al-No} if for every cover $\{ U_\alpha : \alpha \in \Delta \}$ of $A$ by $m$-open sets of $X$, there exists a finite subset $\Delta_0$ of $\Delta$ such that $A \setminus \cup\{ \gamma(U_\alpha) : \alpha \in \Delta_0 \} \in \mathcal{H}$. In this paper, we define and investigate two kinds of strong forms of $\gamma\mathcal{H}$-compact relative to $X$.

Keywords: Hereditary $m$-space, $\gamma\h$-compactness, strong $\gamma\h$-compactness, super $\gamma\h$-compactness

MSC numbers: Primary 54D20, 54D30