$AE(n)$-cover, $X$ is at most $n$-dimensional metric space and $\mathcal{H}$ is closed, locally finite and locally finite-dimensional. By the Carrier Theorem, there exists a map $g \colon X \to Y$ that extends $g_0$ and that is carried by $C \circ D$. This implies that for each $x \in X$ there exists $F \in {{\mathcal{F}_{}}}$ such that $x \in f^{-1}(F)$ and $g(x) \subset p^{-1}(F)$. Therefore $p(g(x)) \in F$ so $p \circ g$ is ${{\mathcal{F}_{}}}$-close to $f$.

A lifting condition for inverse limits
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\[def:limit\] Let $$Z=\lim_{\longleftarrow}\left( Z_1\xleftarrow{p_1}Z_2\xleftarrow{p_2}\cdots \right).$$ Let ${{\mathcal{F}_{i}}}$ be a cover of $Z_i$. We define the following conditions.

1.  For each $i$, $Z_i$ is a complete metric space.

2.  For each $i$, the bonding map $p_i \colon Z_{i+1} \to Z_i$ is surjective and $1$-Lipschitz.

3.  For each $i$, ${{\mathcal{F}_{i}}}$ is either an open cover or a closed, locally finite, locally finite-dimensional cover.

4.  For each $i$, the pull-back $p_i^{-1}({{\mathcal{F}_{i}}})$ is an $AE(n)$-cover.

5.  $\sum_i \operatorname{mesh}{{\mathcal{F}_{i}}} < \infty$.

We denote short projections in the inverse limit by $\pi^k_i \colon Z_k \to Z_i (k > i)$ and long projections in the inverse limit by $\pi_k \colon Z \to Z_k$.

\[lem:limit lift\] Let $$Z=\lim_{\longleftarrow}\left( Z_1\xleftarrow{p_1}Z_2\xleftarrow{p_2}\cdots \right).$$ Let ${{\mathcal{F}_{i}}}$ be a cover of 