$\pi_1\circ g_0:A\to Z_1$ to a map $f_1:U\to Z_1$, where $U$ is an open neighborhood of $A$ in $X$. By Lemma \[lem:limit lift\], $f_1$ can be lifted to a map $g \colon U \to Z$ such that $g_{|A} = g_0$. This is the extension of $f$ we sought.

Local $k$-connectedness of inverse limits of polyhedra
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Let $$\label{eq:sequence}\tag{L}
    X = \lim_{\longleftarrow}\left( K_1 \xleftarrow{p_1} K_2 \xleftarrow{p_2} \cdots.\right)$$ Fix $m$ and let $A \subset K_m$ be a subcomplex of $K_m$. A [[**restriction of to $A$**]{}]{} is the inverse limit $$\label{eq:restriction}\tag{R}
    X' = \lim_{\longleftarrow} \left(K'_m \xleftarrow{p'_m} K'_{m+1} \xleftarrow{p'_{m+1}} \cdots\right),$$ where $K'_m =\ A$ and for each $j \geq m$, $K'_{j+1} = p^{-1}_j(K'_j)$ and $p'_j = p_j \mid_{K'_{j+1}}$.

\[lem:restriction\] Let be the restriction of  to a subcomplex $A \subset K_m$.

If, for each $i$, satisfies the conditions:

1.  $K_i$ is a polyhedron and

2.  $p_i$ is a quasi-simplicial map that is surjective and $n$-regular,

then so does . The inverse limit $X'$ is homeomorphic to $\pi_m^{-1}(A)$, where $\pi_m \colon X \to X_m$ denotes the long projection.

We have $$\pi^{-1}_m(A) = \lim_{\longleftarrow} \left(K'_1 \xleftarrow{p'_1} K'_{2} \xleftarrow{p'_{2}} \cdots\right),$$ where $K'_j = p_j(K'_{j+1})$ and $p'_j = p_j \mid_{K'_{j+1}}$ for $j < m$. The restriction is the same sequence 