into $X$ from a closed subset $A$ of an $n$-dimensional metric space extends over an open neighborhood of $A$. The class of absolute neighborhood extensors in dimension $n$ is denoted by $ANE(n)$ and its elements are called $ANE(n)$-spaces.

We say that a metric space $X$ is an [[**absolute extensor in dimension $n$**]{}]{} if every map into $X$ from a closed subset of an $n$-dimensional metric space $Y$ extends over the entire space $Y$. The class of absolute extensors in dimension $n$ is denoted by $AE(n)$ and its elements are called $AE(n)$-spaces.

\[def:AE(C)\] Let $\mathcal{C}$ be a class of topological spaces. We let $AE(\mathcal{C})$ denote the class of [[**absolute extensors for all spaces from the class $\mathcal{C}$**]{}]{}. We write $AE(X)$ for $AE(\{X\})$.

Absolute extensors and absolute neighborhood extensors in dimension $n$ were characterized by Dugundji in the following theorem.

\[thm:dugundji\] Let $X$ be a metric space. Then,

1.  $X\in ANE(n)\iff X$ is locally $k$-connected for all $k< n$; and

2.  $X\in AE(n)\iff X\in ANE(n)$ and $X$ is $k$-connected for all $k< n$.

\[lem:union is ae\] Assume that $A_1 \subset A_2 \subset \ldots$ is a sequence of subsets of a metric space such that each $A_i$ is closed and for each $i$, 