Let $(X,\leqslant )$ be a poset, we define the upper topology has the principle upper sets, that is upper sets of the form $\left \{ \uparrow x:x\in P\right \} $, as the subbase. We can define Alexandroff topology and Scott topology as well.
Let $Y$ be a subset of $X$. The subspace topology on $Y$ has as opens the subsets of $Y$ of the form $U\cap Y, U$ open in $X$.
The truth is subspace topologies of Alexandroff-discrete spaces are again Alexandroff-discrete.(Easy to prove) and a subspace of a Scott space itself can not be a Scott space.(here)
How to prove the topology of $Y_u$($Y$ with upper topology) is strictly coarser than the one(subspace topology) induced from $X$?
I will be very appreciate if anyone give an example.