A shrinkability criterion for complete metric spaces

A shrinking criterion for upper semicontinuous decompositions of a locally compact metric space is extended to complete metric spaces, answering a question of Daverman (1986). This particular criterion thus fits into the general framework exhibited by the other results in this area (Daverman, 1986, Sections 5 and 6). This includes particularly the fundamental theorem of R.D. Edwards and L.C. Glaser (1972), later polished by A. Marin and Y.M. Visetti (1975) which shows that shrinkability of an upper semicontinuous decomposition of a complete metric space is equivalent to the projection map being a near-homeomorphism.