Let $κ$ be an infinite cardinal, $S$ a set of cardinality $κ$, and let $I = [0, 1]$ be the closed unit interval. Define an equivalence relation $E$ on $I × S$ by $(x,α) E (y,β)$ if either $x = 0 = y$ or $(x,α) = (y,β)$. Let $H(κ)$ be the set of all equivalence classes of $E$; in other words, $H(κ)$ is the quotient set obtained from $I ×S$ by collapsing the subset ${0}×S$ to a point. For each $x ∈ I$ and each $α ∈ S$, $(x,α)$ denotes the element of $H(κ)$ corresponding to $(x,α) ∈ I × S$. The topology induced from the metric $d$ on $H(κ)$ defined by $d((x,α),(y,β))=|x − y|$ if $α = β$, and $d((x,α),(y,β))= x + y$ if $α\not=β$. The set $H(κ)$ with this topology is called the hedgehog of spininess $κ$ and is often denoted by $J(κ)$. The space is a complete, non-compact, metric space of weight $κ$.
A topological space $X$ is an absolute retract for metrizable spaces $(M)$ provided that it is in $(M)$ and is a retract of each space $Y$ in $(M)$ of which it is embedded as a closed subset.
${\bf Question.}$ Is the hedgehog $J(κ)$ of spininess $κ$ an absolute retract ?
