Some Properties of Cells and Gauges

The terminology and notation used in this paper are introduced in the following articles: [20], [25], [2], [7], [18], [21], [8], [3], [4], [16], [13], [23], [14], [17], [5], [11], [12], [1], [19], [6], [10], [15], [22], [24], and [9]. We adopt the following convention: C denotes a simple closed curve, i, j, n denote natural numbers, and p denotes a point of E T . The following propositions are true: (1) BDDC is Bounded. (2) If 〈i, j〉 ∈ the indices of Gauge(C, n) and 〈i + 1, j〉 ∈ the indices of Gauge(C, n), then ρ((Gauge(C, n))1,1, (Gauge(C, n))2,1) = |((Gauge(C, n))i+1,j)1 − ((Gauge(C, n))i,j)1|. (3) If 〈i, j〉 ∈ the indices of Gauge(C, n) and 〈i, j + 1〉 ∈ the indices of Gauge(C, n), then ρ((Gauge(C, n))1,1, (Gauge(C, n))1,2) = |((Gauge(C, n))i,j+1)2 − ((Gauge(C, n))i,j)2|. (4) For every subset S of E T such that S is Bounded holds (proj1)◦S is bounded. (5) Let C1 be a non empty compact subset of E 2 T and C2, S be non empty subsets of E T . If S = C1 ∪ C2 and (proj1) ◦C2 is non empty and lower bounded, then W-boundS = min(W-boundC1,W-boundC2). (6) For every subset X of E T such that p ∈ X and X is Bounded holds W-boundX ¬ p1 and p1 ¬ E-boundX and S-boundX ¬ p2 and p2 ¬ N-boundX. (7) p ∈ WestHalfline p and p ∈ EastHalfline p.