Geodesic
A~Compact Space Is Homotopy Equivalent to a CW-Complex If and Only If It Is an Absolute Neighborhood $h$-Retract
Matematičeskie zametki, Tome 79 (2006) no. 2, pp. 309-310

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It is proved that a compact space is homotopy equivalent to a CW-complex if and only if it is an absolute neighborhood $h$-retract. 
@article{MZM_2006_79_2_a13,
     author = {P. V. Chernikov},
     title = {A~Compact {Space} {Is} {Homotopy} {Equivalent} to a {CW-Complex} {If} and {Only} {If} {It} {Is} an {Absolute} {Neighborhood} $h${-Retract}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {309--310},
     publisher = {mathdoc},
     volume = {79},
     number = {2},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2006_79_2_a13/}
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P. V. Chernikov. A~Compact Space Is Homotopy Equivalent to a CW-Complex If and Only If It Is an Absolute Neighborhood $h$-Retract. Matematičeskie zametki, Tome 79 (2006) no. 2, pp. 309-310. http://geodesic.mathdoc.fr/item/MZM_2006_79_2_a13/