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If/: X->Y is a map of a space X into a space Y, we say that/ is a /oca/ connection in dimension n, provided that for every point yE.Y and every neighborhood N of y there is a neighborhood FCA7' of y such that for 0 g & ^ «-1 any map g : Sk->/-1 F extends to a map g': Bk+1-^f~1N and for any map g: 5"-¡-/"^F the map fg: 5n->F extends to a map fe: Bn+1->N. Using star-refinements of open covers and a standard approximation technique we establish the following theorem (a slightly weaker form of which has been announced by Price [2]).

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ABSTRACT. A theorem is proved which generalizes both the Vietoris-Begle theorem and the cell-like theorem for spaces of finite defomation dimension. The proof is geometric and uses a double mapping cylinder trick.