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The main result is the title theorem asserting that if  G  is any upper semicontinuous decomposition of      S 3 {S^3}       into cell-like sets which is 0-dimensional, in the sense that the image of the nondegenerate elements in      S 3 / G {S^3}/G       is 0-dimensional, then      G ×  S 1 G\, \times \,{S^1}       is shrinkable, and      ( S 3 / G ) ×  S 1 \left ( {{S^3}/G} \right )\, \times \,{S^1}       is homeomorphic to      S 3 ×  S 1 {S^3}\, \times \,{S^1}      .

Cell-like 0-dimensional decompositions of 𝑆³ are 4-manifold factors