I am trying to read "The space of ultrafilters on N convered by nowhere dense sets" by B. Balcar, J. Pelant, P. Simon.
Let $P$ be a dense in itself topological space. Then $n(P)$, the Baire number of $P$, is defined as the minimal cardinality of a collection of nowhere dense subsets of $P$ that covers $P$.
The authors claim that the inequality $\omega_2\leq n(\omega^*)\leq 2^\mathfrak c$ is obvious, but I can't see why. The second inequality holds since every point is a nowhere dense subset and $|\omega^*|=2^\mathfrak c$, but I can't see why the first inequality holds.
Edit: $\omega^*=\mathbb N^*=\beta \mathbb N\setminus \mathbb N$.