In Sierpiński topology the open sets are linearly ordered by set inclusion, i.e. If $S=\{0,1\}$, then the Sierpiński topology on $S$ is the collection $\{ϕ,\{1\},\{0,1\} \}$ such that $$\phi\subset\{0\}\subset\{0,1\}$$ 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 $(\Lambda,\le)$ such that $I$ always contains the void set $ϕ$ and the whole set $X$, i.e. $$I=\{\emptyset,A_\lambda,X:A_\lambda\subset X ,\lambda\in\Lambda\}$$ such that $A_\mu⊂A_\nu$ whenever $\mu\le\nu$.
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?