gn.general topology - How come nowhere dense subsets implies discrete?

Hi, I am reading an article and have encountered a remark in a proof which is not clear to me.
Maybe someone can help?

The proposition is:
Let X be a topological space without isolated points having countable $ pi $-weight and such that every nowhere dense subset in it is closed. Then it is a Pytkeev space.

Here is the begining of the proof:
Let $ x in Cl(A) setminus A$. Then $ x in Cl(Int(Cl(A))) $, because every nowhere dense set is closed (and hence discrete)...

The thing which is not clear to me:
Why can one conclude that every nowhere dense closed set is discrete? Suppose I take the set $ mathbb N$ with the cofinite topology. Then the finite sets are closed and nowhere dense. But as far as I undesrtand they are not discrete since every open set in the topology that contains a finit set also has to contain other points since it is infinite.
Can somone see what am I missing?

The definition of a Ptkeev space:
Let X be a topological space.
A point x is called a Pytkeev point if whenever $ x in overline {Asetminus{x}}$, there exists a countable $ pi $-net of infinite subsets of A. If every point of a space is a Pytkeev point then the space is called a Pytkeev space.

Thanks!