fact that each $f[C]$ is a regular closed set.

Assume now that $C, C'\in \mathcal C $, and let $y\in f[C]\cap f[C']$. We show that $y\notin \operatorname{int}(f[C])$, similarly $y\notin \operatorname{int}(f[C'])$. If toward contradiction $y\in \operatorname{int}(f[C])$, let $O$ be open with $y\in O\subseteq f[C]$. Since $y\in f[C']$ and $f[C']$ is regular closed, there is $y'\in O\cap \operatorname{int}(f[C'])$, so that there exists an open set $V$ with $y'\in V\subseteq f[C]\cap f[C']$. It follows that $ \operatorname{int}(f[C]\cap f[C'])\ne\emptyset$, whence $ \operatorname{int}(C\cap C')\ne\emptyset $, by irreducibility of $f$, and then $ \operatorname{int}(C)\cap \operatorname{int}(C')\ne\emptyset $, against $ \mathcal C $ being a regular quasi-partition.

Recall that we have fixed a fine projective sequence of finite $ \mathcal L_R$-structures $(A_{n}, {\varphi}_{n}^m)$ with projective limit $ \mathbb A $ and with quotient map $p: \mathbb A \to {{\raisebox{.1em}{$ \mathbb A \!$}\left/\raisebox{-.1em}{$R^{ \mathbb A }$}\right.}} $.

\[singletonsdenseiffirreducible\] The following are equivalent:

1.  The set $M$ of points of $ \mathbb A $ whose $R^{ \mathbb A }$-equivalence class is a singleton is dense.

2.  For each $n \in {\mathbb N}$ and $a \in A_{n}$ there are $m>n$ and $b \in A_{m}$ such that if $b' {\mathbin{R}}^{A_m}b$ then ${\varphi}^{m}_{n}(b')=a$.

3.  The quotient map $p: \mathbb A \to {{\raisebox{.1em}{$ \mathbb 