Let us call a space $(X,\tau)$ *totally separated (ts)* if for every two distinct points there is a clopen set containing one, but not the other. If for every topology $\sigma\subseteq\tau$ with $\sigma\neq \tau$ the space $(X,\sigma)$ no longer has this property we call $(X,\tau)$ *minimal ts*.


If $(X,\tau)$ is ts, does $\tau$ contain a topology $\sigma$ such that $(X,\sigma)$ is minimal ts?



Minimal totally separated spaces
First, note that a minimal totally separated space is the same thing as a Stone space. Clearly Stone spaces are minimal totally separated (any coarser topology cannot even be Hausdorff); conversely suppose $X$ is totally separated and not Stone. We may assume the topology on $X$ is generated by its clopen sets (otherwise they generate a coarser totally separated topology). Then $X$ is canonically a dense subspace of the Stone space $S(B)$ of its clopen algebra $B$. If $X$ is not all of $S(B)$, let $u\in S(B)\setminus X$ and $x\in X$. Let $T$ be the quotient of $S(B)$ obtained by identifying $x$ and $u$; the composition $X\to S(B)\to T$ is then injective and induces another totally separated topology on $X$. This new topology is strictly coarser than the original topology: there is some net $(x\_i)$ in $X$ that converges to $u$ in $S(B)$, and this net (which had no limit in $X$ in the old topology) converges to $x$ in the new topology.


Thus a minimal totally separating topology contained in a given topology on $X$ is equivalent to a continuous bijection $X\to S$ from $X$ to a Stone space $S$. If $A$ is the clopen algebra of $S$, then $A$ is naturally a subalgebra of the clopen algebra $B$ of $X$, and the map $X\to S$ is determined by the inclusion $A\to B$. Thus the question can be recast as follows: given a totally separated space $X$ with clopen algebra $B$, is there a subalgebra $A\subseteq B$ such that the canonical map $X\to S(A)$ to the Stone space of $A$ is a bijection?


Let $D$ be any infinite discrete space; then I claim we can find a counterexample $X$ which is a dense subspace of the Stone-Cech compactification $\beta D$. To find such an $X$, note that $|\beta D|=2^{2^{|D|}}$, which is the same as the number of subalgebras of the power set algebra $\mathcal{P}(D)$. We can thus by transfinite induction build a subset $X\subset \beta D$ that contains $D$ and avoids bijecting onto $S(A)$ for each subalgebra $A\subseteq \mathcal{P}(D)$, identifying $\mathcal{P}(D)$ with the clopen algebra of $\beta D$. Since $X$ contains $D$ as a dense subset, a clopen subset of $X$ is determined by its intersection with $D$, and so $\mathcal{P}(D)$ is also the clopen algebra of $X$. By construction, then, the map $X\to S(A)$ is not a bijection for any subalgebra $A$ of the clopen algebra of $X$.


This is, of course, horribly nonconstructive. It would be interesting to see an explicit counterexample.


