RENDICONTI DEL SEMINARIO MATEMATICO

152 D. Papini - F. ZanolinFigure 14: Examples of ̂M ⊆ ĥN and of ̂M ⊆ v̂N (the left and the right figures,respectively). The painted areas represent ̂M as embedded in ̂N. The contours of [ ·] − -sets for the oriented cells ̂M and ̂N are indicated with a bold line.DEFINITION 7. Let Â, ̂B and ̂M be three rectangular cells with A,B,M subspacesof the same topological space X and suppose that M ⊆ A ∩ B.We say that ̂B crosses  in ̂M and writêM ∈ { ⋔ ̂B},if̂M ⊆ h  and ̂M ⊆ v ̂B.4.2. ApplicationsAt this step, we can just reconsider the same main results from [59, 60] already provedfor the stretching property in the case of generalized two-dimensional cells and extendthem to (1, N − 1)-rectangular cells. For instance, we have the following (compare toTheorem 4).THEOREM 12. Suppose that  = (A,A − ) and ̂B = (B,B − ) are oriented cellsin X. If (D,ψ) : Â⊳̂B and there are k ≥ 2 oriented cells ̂M 1 ..., ̂M k such that̂M i ∈ { ⋔ ̂B}, for i = 1,...,k,withM i ∩ M j ∩ D = ∅, for all i ̸= j, with i, j ∈ {1,...,k},