**Motivation:** Let $(X,\tau)$ be a topological space. Then the set $\beta X$ of ultrafilters on $X$ admits a natural topology (cf. Example 5.14 in Adámek and Sousa - D-ultrafilters and their monads), giving rise to a functor $\beta: \operatorname{Top} \to \operatorname{Top}$ which admits the structure of a monad. It turns out that the algebras for this monad, which I'll call "$\beta$-spaces", admit the following description (which one can alternatively take as a definition).

**Definition:** A *$\beta$-space* consists of a topological space $(X,\tau)$ equipped with an additional topology $\tau^\xi$ on $X$ such that

- $(X, \tau^\xi)$ is compact Hausdorff;
- The topology $\tau^\xi$ refines the topology $\tau$; and
- For every $x \in X$ and every $\tau$-open neighborhood $U$ of $x$, there exists a $\tau$-open neighborhood $V$ of $x$ such that the $\tau^\xi$-closure of $V$ is contained in $U$.

**Notes:**

- From (1) and (2) it follows that $(X,\tau)$ is compact.
So if $(X,\tau)$ is additionally Hausdorff, then it admits a unique $\beta$-space structure, namely the one with $\tau^\xi = \tau$ (since continuous bijections of compact Hausdorff spaces are homeomorphisms).

$(X,\tau)$ need not be Hausdorff—e.g., if $\tau$ is the indiscrete topology, then the topology $\tau^\xi$ can be an arbitrary compact Hausdorff topology.

The compact Hausdorff topology $\tau^\xi$ traces back to Manes' theorem, which says that the algebras for the ultrafilter monad on $\operatorname{Set}$ rather than $\operatorname{Top}$ are precisely the compact Hausdorff spaces.

**Questions:**

Are there additional restrictions on the topology $(X,\tau)$ such that it admits a refinement $\tau^\xi$ satisfying (1), (2), (3) (i.e. constituting a $\beta$-space), beyond the fact, as noted, that $X$ must be compact?

Do $\beta$-spaces already have some other name? Or at least, is condition (3) above, relating a topology $\tau$ to a refinement $\tau^\xi$, something which has a name?