Properties of the Internal Approximation of Jordan’s Curve

The articles [19], [25], [14], [10], [1], [16], [2], [3], [24], [11], [18], [9], [26], [6], [17], [7], [8], [12], [13], [20], [15], [4], [5], [21], [23], and [22] provide the notation and terminology for this paper. One can prove the following propositions: (1) For every non constant standard special circular sequence f holds BDD L̃(f) = RightComp(f) or BDD L̃(f) = LeftComp(f). (2) For every non constant standard special circular sequence f holds UBD L̃(f) = RightComp(f) or UBD L̃(f) = LeftComp(f). (3) Let G be a Go-board, f be a finite sequence of elements of E T, and k be a natural number. Suppose 1 ¬ k and k + 1 ¬ len f and f is a sequence which elements belong to G. Then left cell(f, k,G) is closed. (4) Let G be a Go-board, p be a point of E T, and i, j be natural numbers. Suppose 1 ¬ i and i + 1 ¬ lenG and 1 ¬ j and j + 1 ¬ widthG. Then p ∈ Int cell(G, i, j) if and only if the following conditions are satisfied: (i) (G ◦ (i, j))1 < p1, (ii) p1 < (G ◦ (i + 1, j))1, (iii) (G ◦ (i, j))2 < p2, and (iv) p2 < (G ◦ (i, j + 1))2. (5) For every non constant standard special circular sequence f holds BDD L̃(f) is connected. Let f be a non constant standard special circular sequence. Observe that BDD L̃(f) is connected.