Reversible filters

A space is reversible if every continuous bijection of the space onto itself is a homeomorphism. In this paper we study the question of which countable spaces with a unique non-isolated point are reversible. By Stone duality, these spaces correspond to closed subsets in the \v{C}ech-Stone compactification of the natural numbers $\beta\omega$. From this, the following natural problem arises: given a space $X$ that is embeddable in $\beta\omega$, is it possible to embed $X$ in such a way that the associated filter of neighborhoods defines a reversible (or non-reversible) space? We give the solution to this problem in some cases. It is specially interesting whether the image of the required embedding is a weak $P$-set.