Is $\beta\omega$ hereditarily irresolvable?

Question

$X$ is called resolvable if it can be represented as a union of two disjoint dense sets, it is irresolvable otherwise. Moreover it is hereditarily irresolvable (HI) if every subspace of X is irresolvable. My question is whether the space $\beta\omega$ is HI? I could not find the right way to approach this question.

Answer 1

It seems like very few spaces are hereditarily irresolvable. For example if $X$ is Hausdorff then for any $a,b \in X$ the doubleton $\{a,b\}$ is a discrete subspace and so has no proper dense subspaces. We conclude Hausdorff spaces are not hereditarily irresolvable.

Therefore $\beta \omega$ is not hereditary irresolvable. Indeed $\beta \omega$ is resolvable. To see this let $A \subset \beta \omega$ be the set of ultrafilters of some fixed nontrivial type and $B = \beta \omega - A$. Then $A$ and $B$ are dense, disjoint, and union to the whole space.

Edit: This answer makes very little sense.