distinct and $\neg({\varphi}_{m}(x) {\mathbin{R}}^{A_m}{\varphi}_{m}(y))$, so $d_{R^{A_m}}({\varphi}_{m}(x), {\varphi}_{m}(y)) = 2$. Therefore the property does not hold for ${\varphi}_n(x), {\varphi}_n(y)$.

\[rconnectedness\] Let $A$ be a topological $\mathcal L_{R}$-structure and $B \subseteq A$. We say $B$ is $R$-*connected* if for any two clopen sets $U, U' \subseteq A$ such that $U\cap B, U'\cap B$ partition $B$, there are $x \in U\cap B, x' \in U' \cap B$ such that $x {\mathbin{R}}^Ax'$.

Notice that if $A$ is a finite $\mathcal L_{R}$-structure and $R^A$ is symmetric, $R$-connectedness coincides with the usual notion of connectedness for the graph $R^{A}$.

\[oldparagraph\] Let $A$ be a prespace. Then the image of an $R$-connected closed subset $B \subseteq A $ under the quotient map $p: A \to  {{\raisebox{.1em}{$A \!$}\left/\raisebox{-.1em}{$R^{A }$}\right.}}$ is closed and connected.

The set $p[B]$ is closed as $p$ is a closed map. If $p[B]$ were disconnected, let $C, C'$ be disjoint, nonempty, closed subsets of $ {{\raisebox{.1em}{$A \!$}\left/\raisebox{-.1em}{$R^{A }$}\right.}} $ such that $p[B]=C\cup C'$. Then $p^{-1}(C)\cap B, p^{-1}(C')\cap B$ are disjoint, nonempty, closed subsets of $A $ whose union is $B$. Let $U, U'$ be disjoint clopen subsets of $A $ with $p^{-1}(C)\cap B\subseteq U, p^{-1}(C')\cap B\subseteq U'$. By the assumption, there are $u\in p^{-1}(C)\cap B, 