Which topological spaces does every ultrafilter on the clopens converge?

Question

Compact spaces are exactly those in which every ultrafilter on $\mathcal{P}X$ converges, or equivalently ultrafilters on the frame of opens $\mathcal{O}X$. In Stone spaces, or more generally compact $0$-dimensional spaces, since there is a basis of clopens, any ultrafilter on $\mathcal{B}X$ the boolean algebra of clopens will also converge.

But it seems that every ultrafilter on $\mathcal{B}X$ converging does not imply $0-$dimensionality, as the Sierpinski space has boolean algebra $2$, and its only ultrafilter is the set {$S$}, but this does converge to $0$, but the Sierpinski space does not have a basis of clopens.

Therefore I am interested to know if there is a characterisation of for which spaces all ultrafilters on the boolean algebras of clopens will converge?

Answer 1

These are the $H(i)$-spaces studied in

C. Scarborough, A. Stone, Products of Nearly Compact Spaces, Trans. Amer. Math. Soc., 124 (1966), 131-147.

A Hausdorff space $X$ has the property that each of its open ultrafilters converges if and only if $X$ is H-closed (i.e. a closed subspace of every other Hausdorff space in which it embeds). A regular Hausdorff space has this property if and only if it is compact.

Nevertheless, the $H(i)$-property studied by Scarborough and Stone was introduced purposefully to avoid separation assumptions. It is remarked in the paper that;

A space $X$ is an $H(i)$-space if and only if every open cover of $X$ has a finite subfamily whose union is a dense subspace of $X$.

There are also other classes of spaces considered in the paper. Namely the $H(ii)$-space, the $R(i)$-spaces, and the $R(ii)$-spaces.