If Y is a subset of X, then Y inherits a total order from X. The set Y therefore has an order topology, the induced order topology
What does it mean "inherits" ? Is this a a Kripke model M construction?
Because inhabited sets are the same as nonempty sets in classical logic, it is not possible to produce a model in the classical sense that contains a nonempty set X but does not satisfy "X is inhabited". But it is possible to construct a Kripke model M that satisfies "X is nonempty" without satisfying "X is inhabited".
Is a subset of total order ?
In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation
or is a cover set ?
a cover of a set $X$ is a collection of sets whose union contains $X$ as a subset
i don't understand that word "inherits", for me is cryptic
Having a total order on $X$ says that every pair of elements of $X$ are comparable and the order is transitive. If you extract a subset $Y$, between any two elements you can use the comparison in the total order on $X$. It will be transitive, so it is a total order on $Y$. "Inherits" means that given you have a total order on $X$ you also have one on $Y$ that comes from the one on $X$.