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 {A\setminus{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!