In this paper we investigate further the results given in [7]. Let $X$ be a topological space and $\mathcal K(X)\subset\exp(X)$ the space of all compact subsets of $X$ with the finite topology. The main results in section 2 are: (a) In the class of spaces $\{X\}$ where $\mathcal K(X)$ is normal, the notions of hypercountably compactness and strongly countably compactness coincide, (b) In the class of spaces $\{X\}$ where $\exp(X)$ satisfies the first axiom of countability, the notions of compactness and hypercountably compactness coincide.