An alternative proof of Cauty–Geogheganʹs characterization of ANRs

In 1994, R. Cauty proved Geogheganʹs conjecture on ANRs that a metrizable space X is an ANR if and only if every open set in X has the homotopy type of a CW-complex. In this note, by using the mapping cylinder technique, we provide an alternative short proof of this characterization of ANRs.