no assumption of local finiteness of complexes in the statement of Theorem \[thm:main theorem\].

It is known that (under suitable assumptions) if the bonding maps in the inverse limit are $n$-soft, then the inverse limit is $k$-connected for $k < n$, [@chigogidze1989]. The condition for local $k$-connectedness given in the present paper may be regarded as a combinatorial analog of this statement for inverse limits of polyhedra. Note that $n$-regular bonding maps need not be $n$-soft.

Preliminaries
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In this section we set the basic definitions and reference known results that will be used in the later sections.

Absolute extensors in dimension $n$
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We say that a space $X$ is [[**$k$-connected**]{}]{} if each map $\varphi \colon S^k \to X$ from a $k$-dimensional sphere into $X$ is null-homotopic in $X$.

We say that a space $X$ is [[**locally $k$-connected**]{}]{} if for each point $x \in X$ and each open neighborhood $U \subset X$ of $x$ there exists an open neighborhood $V$ of $x$ such that each map $\varphi \colon S^k \to V$ from a $k$-dimensional sphere into $V$ is null-homotopic in $U$.

We say that a metric space $X$ is an [[**absolute neighborhood extensor in dimension $n$**]{}]{} if every map 