u'\in p^{-1}(C')\cap B$ with $u{\mathbin{R}}^{A }u'$, contradicting the disjointness of $C, C'$.

For the remainder of the section we fix 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.}} $.

\[generalprop\]

1.  The mesh of the sequence $\left(\setnew{ {\varphi}_n^{-1}(a)}{a \in A_{n}} \right)_{n \in {\mathbb N}}$ tends to $0$. In particular, the sets $ {\varphi}_n^{-1}(a)$ for $n\in {\mathbb N}, a\in A_n$ form a basis for the topology of $ \mathbb A $.

2.  The mesh of the sequence $\left(\setnew{p[ {\varphi}_n^{-1}(a)]}{a \in A_{n}} \right)_{n \in {\mathbb N}}$ tends to $0$.

\(1) Suppose that there is $\varepsilon > 0$ such that for infinitely many $n \in {\mathbb N}$, there is $a_{n} \in A_{n}$ with $\operatorname{diam}( {\varphi}_n^{-1}(a_n))\ge \varepsilon$. Consider the forest $T= \setnew{ {\varphi}_{n'}^n(a_n)}{n'<n} $, so that $ \operatorname{diam}( {\varphi}_n^{-1}(b))\ge\varepsilon $ for every $b$ in the forest, if $b\in A_n$. Let $u=(b_0, b_1, \ldots )\in \mathbb A $ be an infinite branch in $T$. Since $$n<n'\Rightarrow {\varphi}_{{n'}}^{-1}(b_{n'}) \subseteq {\varphi}_{n}^{-1}(b_{n})$$ it follows that the sequence $ {\varphi}_n^{-1}(b_n)$ converges in $ \operatorname{\mathcal K}( \mathbb A )$ to $K=\bigcap_{n\in {\mathbb N}} {\varphi}_n^{-1}(b_n)$ with 