I am new to the ultrafilters, so I apologise if the question is too elementary.
Let S be a collection of sets with the finite intersection property, in a non-compact Hausdorff space. S can be extended to an ultrafilter but can it be extended to a convergent ultrafilter, i.e., one containing the neighbourhood filter of some point?
This becomes simple if S has non-empty intersection (but this is not known) or when the space is compact (but it is not). If this can not be shown in general then, perhaps, under some more conditions on the space or the sets in S?
I would appreciate references to relevant techniques/literature as much as a direct answer.