Some Properties of Cells and Arcs

The articles [29], [36], [35], [14], [33], [1], [28], [31], [37], [4], [3], [34], [5], [15], [30], [27], [10], [18], [13], [2], [25], [26], [22], [9], [11], [7], [32], [17], [12], [21], [19], [20], [6], [24], [23], [8], and [16] provide the notation and terminology for this paper. For simplicity, we adopt the following convention: E denotes a compact non vertical non horizontal subset of E T, C denotes a compact connected non vertical non horizontal subset of E T, G denotes a Go-board, i, j, m, n denote natural numbers, and p denotes a point of E T. Let us observe that every simple closed curve is non vertical and non horizontal. Let T be a non empty topological space. Note that there exists a union of components of T which is non empty. Next we state a number of propositions: