Let V be open in Rn and V Vj jN, W Wk kN be coverings of V If j jN is a Cp partition of unity on V subordinate to V and k kN is a Cp partition of unity on V subordinate to W, prove that j k)j,kN is a Cp partition of unity on V subordinate to the covering Vj Wk j,kN ...

Clearly j k belongs to C p c R n j k 0 and spt j k V j W k Also Given a compact su ...

Let V be open in Rn and V = {Vj}jN, W = {Wk}kN be coverings of V.

Question:

Let V be open in Rn and V = {Vj}j∈N, W = {Wk}k∈N be coverings of V. If {ɸj}j∈N is a Cp partition of unity on V subordinate to V and {ψk}k∈N is a Cp partition of unity on V subordinate to W, prove that {ɸj ψk)j,k∈N is a Cp partition of unity on V subordinate to the covering {Vj ∩ Wk}j,k∈N-