In this paper, we shall show: (1) Properties $[\mathfrak{a,b}]^{r}$-compactness, $[\mathfrak{a,b}]^{r}$-refinability and weakly $[\mathfrak{a,b}]^{r}$-refinability are preserved under taking countable closed sums, $F_{\sigma}$-subsets and preimages of perfect mappings. \par (2) (GCH) Let $X$ be a space with $t(X) \le \mathfrak{n}$ and $Y$ be a bounded $\mathfrak{n}$-compact space for some cardinal $\mathfrak{n}$. If $X$ is $[\mathfrak{a,b}]^{r}$-compact (resp. $[\mathfrak{a,b}]^{r}$-refinable, weakly $[\mathfrak{a,b}]^{r}$-refinable) and $L(Y) < \mathfrak{a}$, then $X \times Y$ is $[\mathfrak{a,b}]^{r}$-compact (resp. $[\mathfrak{a,b}]^{r}$-refinable, weakly $[\mathfrak{a,b}]^{r}$-refinable). \par (3) Suppose that $\mathfrak{a}$ is a regular cardinal with $\mathfrak{a} \ge \omega_{1}$. Let $X$ be a separable metric space and $Y$ be a $P(\omega)$-space. If $Y$ is $[\mathfrak{a,b}]^{r}$-compact (resp. $[\mathfrak{a,b}]^{r}$-refinable, weakly $[\mathfrak{a,b}]^{r}$-refinable), then $X \times Y$ is $[\mathfrak{a,b}]^{r}$-compact (resp. $[\mathfrak{a,b}]^{r}$-refinable, weakly $[\mathfrak{a,b}]^{r}$-refinable).