Some Properties of Cells and Arcs
Robert Milewski, Andrzej Trybulec, Adam Naumowicz · 2007
The notation and terminology used in this paper are introduced in the following papers: [25], [2], [11], [26], [21], [12], [3], [5], [30], [7], [28], [6], [18], [22], [17], [24], [20], [23], [8], [10], [16], [1], [27], [9], [4], [15], [32], [19], [29], [31], [13], and [14]. For simplicity, we adopt the following convention: E denotes a compact non vertical non horizontal subset of E2 T, C denotes a compact connected non vertical non horizontal subset of E2 T, G denotes a Go-board, i, j, m, n denote natural numbers, and p denotes a point of E2 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. The following propositions are true: (1) Let T be a non empty topological space and A be a non empty union of components of T . If A is connected, then A is a component of T . (2) For every finite sequence f holds f is empty iff Rev(f) is empty. (3) Let D be a non empty set, f be a finite sequence of elements of D, and given i, j. If 1 ¬ i and i ¬ len f and 1 ¬ j and j ¬ len f, then mid(f, i, j) is non empty. (4) Let f be a non empty finite sequence of elements of E2 T and p be a point of E2 T. If 1 ¬ len f and p ∈ L(f), then (⇂ f, p)(1) = f(1). (5) Let f be a non empty finite sequence of elements of E2 T and p be a point of E2 T. If f is a special sequence and p ∈ L(f), then (⇃ p, f)(len ⇃ p, f) = f(len f).