RENDICONTI DEL SEMINARIO MATEMATICO

150 D. Papini - F. ZanolinFigure 13: A possible picture of a (1, 2)-rectangular cell ̂N, where we have put inevidence the two components of the N − set which are painted with a darker color.Our definition of (1, N − 1)-rectangular cell is borrowed from that of h-set given byZgliczyński and Gidea [82, Definition 1] and considered also by Pireddu and Zanolinin [61]. However, we point out that, differently than in [82] and [61], we don’t assumehere N to be a subset of R N and moreover in the present case the homeomorphism c Nis defined only on N whence in the above cited articles c N was defined on the wholespace X. We also define the setsN − l:= c N −1 ({−1} × [−1, 1] N−1 ), N − r := c N −1 ({1} × [−1, 1] N−1 ),conventionally called the left and the right faces of ̂N, as well as the setIf we define nowN − := N − l∪ N − r .Ñ := (N,N − ),we have that Ñ is a path-oriented spaces which possesses the FPP-γ.Our definition of oriented cell ̂N fits with that of (1, N − 1)-window considered byGidea and Robinson in [25] and, in the special case N = 2, is equivalent to that oftwo-dimensional oriented cell by Papini and Zanolin in [60].For completeness we also recall the form that the stretching condition takes withrespect to the path-oriented spaces determined by the rectangular cells that we have justdefined.Let  = (A, c A ; X, N 1 ) and ̂B = (B, c B ; Y, N 2 ) be two rectangular cells containedin the Hausdorff topological spaces X and Y, respectively. Let φ : X ⊇ D φ → Y bea map (not necessarily continuous on its whole domain D φ ) and let us consider a setD ⊆ D φ .