A note on $\delta$-strongly compact cardinals

In this paper we investigate more characterizations and applications of $\delta$-strongly compact cardinals. We show that, for a cardinal $\kappa$ the following are equivalent: (1) $\kappa$ is $\delta$-strongly compact, (2) For every regular $\lambda \ge \kappa$ there is a $\delta$-complete uniform ultrafilter over $\lambda$, and (3) Every product space of $\delta$-Lindel\"of spaces is $\kappa$-Lindel\"of. We also prove that in the Cohen forcing extension, the least $\omega_1$-strongly compact cardinal is a precise upper bound on the tightness of the products of two countably tight spaces.