is either empty or belongs to $\mathcal{C}$.

We say that a cover is [[**locally finite-dimensional**]{}]{} if its nerve is locally finite-dimensional.

A [[**carrier**]{}]{} is a function $C \colon {{\mathcal{F}_{}}} \to {{\mathcal{G}_{}}}$ from a cover ${{\mathcal{F}_{}}}$ of a space $X$ into a collection ${{\mathcal{G}_{}}}$ of subsets of a topological space such that for each $\mathcal{A} \subset {{\mathcal{F}_{}}}$ if $\bigcap_{A\in\mathcal{A}}A \neq \emptyset$, then $\bigcap_{A \in
  \mathcal{A}} C(A) \neq \emptyset$. We say that a map $f$ is [[**carried by $C$**]{}]{} if it is defined on a closed subset of $X$ and $f(F) \subset C(F)$ for each $F \in {{\mathcal{F}_{}}}$. Here we write $f(F)$ to mean $f(F\cap \operatorname{dom}(f)).$

Assume that $C \colon {{\mathcal{F}_{}}} \to {{\mathcal{G}_{}}}$ is a carrier such that ${{\mathcal{F}_{}}}$ is a cover of a space $X$ and ${{\mathcal{G}_{}}}$ is an $AE(X)$-cover of another space. If ${{\mathcal{F}_{}}}$ is closed, locally finite, and locally finite-dimensional, then each map carried by $C$ extends to a map of the entire space $X$, also carried by $C$.

If ${{\mathcal{F}_{}}}$ is a closed locally finite locally finite-dimensional $AE(n)$-cover of a space $Y$, then any two ${{\mathcal{F}_{}}}$-close maps from a metric space of dimension less than $n$ into $Y$ are ${{\mathcal{F}_{}}}$-homotopic.

Covers
------

We regard covers as indexed collections 