One Characterisation of Absolute Retracts and its Applications

We prove that $ANR(\mathfrak{M}) \cap AR_{\varepsilon} (\mathfrak{M}) = AR(\mathfrak{M})$ and some propositions related to the following problem: if $X$ is a metric compacta, does the condition $exp_2 X \in AR$ imply $X \in AR_\epsilon(\mathfrak{M})$.