$B_{n}$. But $B_{n}$ is $R$-connected, so there are $a \in B_{n} \cap {\varphi}_n[U], a' \in B_{n} \cap {\varphi}_n[U']$ such that $a {\mathbin{R}}^{A_n}a'$, and thus there exist $x \in {\varphi}_n^{-1}(a) \subseteq U, x' \in {\varphi}_n^{-1}(a') \subseteq U'$ such that $x {\mathbin{R}}^{ \mathbb A }x'$. So $K$ is $R$-connected.

\[quotlimitconnected\] If $B_{n} \subseteq A_{n}$ are $R$-connected subsets and $(p[{\varphi}^{-1}_{n}(B_{n})])_{n \in {\mathbb N}}$ converges in $\operatorname{\mathcal K}( {{\raisebox{.1em}{$ \mathbb A \!$}\left/\raisebox{-.1em}{$ R^{ \mathbb A }$}\right.}} )$ to some $K$, then $K$ is connected.

Let $n_k$ be an increasing sequence of natural numbers such that $ {\varphi}_{n_k}^{-1}(B_{n_k})$ converges in $ \operatorname{\mathcal K}( \mathbb A )$, say $\lim_{k\rightarrow\infty } {\varphi}_{n_k}^{-1}(B_{n_k})=L$. Then $$\lim_{n\rightarrow\infty }p[ {\varphi}_n^{-1}(B_n)]= \lim_{k\rightarrow\infty }p[ {\varphi}_{n_k}^{-1}(B_{n_k})]=p[L],$$ whence $K=p[L]$. Now apply .

Irreducible functions and regular quasi-partitions
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Given topological spaces $X$, $Y$, a continuous map $f:X\to Y$ is *irreducible* if $f[K] \neq Y$ for all proper closed subsets $K \subset X$.

We recall some basic results on irreducible closed surjective maps between compact metrizable spaces, whose proofs can be found in [@MR785749]. Let $f:X\to Y$ be such a map. Given $A\subseteq X$, let $f^{\#}(A) = \setnew{y\in Y}{f^{-1}(y) \subseteq A}$. If $O\subseteq X$ is an open set, then $f^{\#}(O)$ is open and $f^{-1}(f^{\#}(O))$ is dense 