In this answer of Andreas Blass to a question of mine, it is said that every infinite closed subset of $βℕ$ contains a copy of $βℕ$. I have a proof, but I think it can be improved and would like to know if there is a more standard or general way of seeing that. Is there any reference? I didn't find that in Andreas Blass' article.
Proof. First, we show that if $X$ is any infinite Stone space, then there is an embedding of the discrete space $ℕ$ inside of $X$. Consider any point $x_0 ∈ X$. If there is some clopen subset $U$ containing $x_0$ and such that $X\setminus U$ is infinite, we continue the induction after replacing $X$ with $X\setminus U$. If this isn't possible, it means that $X$ is the one-point compactification of $X\setminus\{x\}$ and we can take any injection $ℕ→X\setminus\{x\}$. Suppose now that $X ⊆ βℕ$ is a closed subspace and consider an embedding $ι : ℕ↪X$. We construct inductively disjoint subsets $P_i ⊆ ℕ$ for each $i∈ℕ$ such that $x_i ∈ \overline{P_i}$. We show that the map $βℕ→X⊆βℕ$ induced by $ι$ is injective. By Stone duality, it means that for each subset $U ⊆ ℕ$, there is some subset $V ⊆ ℕ$ such that the inverse image of $\overline{V}$ intersected with $ℕ$ is $U$. We can take $V = \bigcup_{i∈U} P_i$.
More generally, this shows that if $X$ is a closed subspace of a Stonean space and if $ℕ→X$ is an embedding in $X$, with the discrete topology on $ℕ$, then $βℕ→X$ is an embedding. But I don't think that Stonean spaces are stable under taking closed subspaces (anyone has a counter-example?). Is there an invariant of Stone spaces stable under taking closed subspaces ensuring this property? The property itself is stable under closed subspaces but is there something that allows the reasoning itself to be reproduced inside of the space $X$ without relying on a bigger space?