On the generalized Sierpinski space
In Sierpiński topology the open sets are linearly ordered by set inclusion, i.e. If , then the Sierpiński topology on S is the collection such that we can generalize it by defining a topology analogous to Sierpiński topology with nested open sets on any arbitrary non-empty set as follows: Let X be a non-empty set and I a collection of some nested subsets of X indexed by a linearly ordered set such that I always contains the void set and the whole set X, i.e.
such that whenever .
Then it is easy to show that I qualifies as a topology on X.
My questions are:
(1) Is there a name for such a topology in general topology literature?
(2) Is there any research paper studying such type of compact, non-Hausdorff and connected chain topologies?