Remark 3

Let V be a manifold of class C^r. Let M be a subset of V and x and y elements of M. Consider the following property:

P_{M, x, y}: there exist I, x_0, x_1, \ldots, x_n, f_1, \ldots, f_n such that: (a) I is a connected open subset of \mathbf{K}; (b) x_0, \ldots, x_n are in M, x_0 = x, x_n = y; (c) for 1 \leq i \leq n, f_i is a mapping of class C^r of I into V which takes the values x_{i-1} and x_i and f_i(I) \subset M.

We shall say that M is a C^r-connected subset of V if, for all elements x, y of M, the property P_{M, x, y} holds.