Absolute neighborhood retract

Remarks

Some authors define these classes for compact metrizable spaces only, or for other classes of spaces; the definition above is for metrizable spaces. Equivalently, $X$ is an ANR if every continuous map $f \colon A \to X$ from a closed subset $A$ of a metrizable space $Y$ extends to a continuous map on some open set $U \supseteq A$.

Example: $X = [0, 1]$ is an AR. Given $h$ as above, the map $h^{-1} \colon h(X) \to [0, 1]$ is continuous on the closed set $h(X)$, and by the Tietze extension theorem it extends to a continuous $g \colon Y \to [0, 1]$. Then $r = h \circ g$ is a retraction of $Y$ onto $h(X)$.

Non-example: $X = \set{0} \cup \set{1/n : n \ge 1}$, a closed subset of $Y = \R$, with $h$ the inclusion. Any open $U \supseteq X$ contains an interval $(-\varepsilon, \varepsilon)$. A retraction $r \colon U \to X$ would map this connected interval onto a connected subset of $X$ that contains $0$ and $1/n$ for large $n$, which is impossible.