Rectifiably ambiguous points of planar sets

Abstract

Denote by P the Euclidean plane with a rectangular Cartesian coordinate system where the x-axis is horizontal and the y-axis is vertical. An arc in P shall mean a simple continuous curve Λ:{t: 0 ≦ t < 1} P having the properties that limitt→1 Λ(t) exists and limitt→1Λ(t) ≠ Λ(t0) for 0 ≦t0 < 1. An arc at a pointζ in P shall be an arc Λ where limt→1 Λ(t) = ζ. If S is an arbitrary subset of the plane, ζ is termed an ambiguous point relative to S provided there are arcs Λ and Γ at ζ with Λ ⊆ S and Γ ⊆ P–S; such arcs are referred to as arcs of ambiguity at ζ. If A is a set of arcs we say a point ζ in P is accessible via A provided there is an arc at ζ which is an element of A. If B is also a collection of arcs, then A and B are said to be pointwise disjoint if whenever α∈A and β ∈ B, α ∩ β = Ø. The collections A and B are said to be terminally arcwise disjoint if whenever α ∈ A and β ∈ B and both α and β are arcs at a point ζ in P, then a ∩ β contains no arc at ζ. If S is a planar set, we let A(S) denote the set of all arcs contained in S. Note that if S ∩ T = Ø then A(S) and A(T) are pointwise disjoint collections of arcs.