The Lattice of Domains of an Extremally Disconnected Space by Zbigniew Karno. One can verify that there exists a topological space which is discrete, anti-discrete, strict, and non empty.
X is discrete iff every subset of X is closed.
Let X be a discrete non empty topological space. Observe that every subspace of X is open, closed, and discrete. Let X be a discrete non empty topological space. Note that there exists a subspace of X which is discrete and strict. Then X is anti-discrete.