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 ?



Hedgehog of spininess $κ$ is an absolute retract?
The answer is positive because the hedgehog $J(κ)$ is $AE$.


