  For a tensor triangulated category and any regular cardinal $\alpha$ we study
the frame of $\alpha$-localizing tensor ideals and its associated space of
points. For a well-generated category and its frame of localizing tensor ideals
we provide conditions such that the associated space is obtained by refining
the topology of the corresponding space for the triangulated subcategory of
$\alpha$-compact objects. This is illustrated by several known examples for
$\alpha=\aleph_0$, and new spaces arise for $\alpha>\aleph_0$.
