Rectifiably ambiguous points of planar sets

Frederick Bagemihl, Paul D. Humke
1975 Journal of the Australian Mathematical Society  
Communicated by G. Szekeres 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 A:{t:0 ^ t < 1} -> P having the properties that limit^jA^) exists and limit,., t A(V) # A(f 0 ) for 0 ^ t 0 < 1. An arc at a point £ in P shall be an arc A where l i m ,^ A(f) = £. If S is an arbitrary subset of the plane, £ is termed an ambiguous point relative to S provided there
more » ... e arcs A and F at £ with A^S and F ^ P-S; such arcs are referred to as arcs of ambiguity at £. If y4 is a set of arcs we say a point £ in P is accessible via ^4 provided there is an arc at £ 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 oceA and fieB, a n / ? = 0 . The collections /I and 5 are said to be terminally arcwise disjoint if whenever aeA and fleB and both a and /? are arcs at a point £ in P , then a n /? contains no arc at £. If S is a planar set, we let s#{S) denote the set of all arcs contained in S. Note that if S n T -0 then s#(S) and J&(T) are pointwise disjoint collections of arcs. In this paper we deal with accessibility of points via sets of rectifiable arcs and sets of totally nonrectifiable arcs, and related questions in ambiguous point theory. (An arc a is totally nonrectifiable if a/[*i, ? 2 ] is nonrectifiable for 0 ^ t x < t 2 ^ 1 •) Let M denote the set of all planar rectifiable arcs, and let J/d enote the set of all planar totally nonrectifiable arcs. Bagemihl (1966) showed that there is a set S x such that every point of the plane is an ambiguous point relative to S x and the arcs of ambiguity may be chosen to be rectifiable. In the first part of this paper we strengthen this result by showing that both 1. 2. Secondly, we use Si to define a set S 2 s= P such that every point of the plane is an ambiguous point relative to S 2 and both 1. s/(S 2 ) <= <V, 2. ^(P-S 2 )<=^T. 85 use, available at https://doi.
doi:10.1017/s1446788700023971 fatcat:bt2rnouihjfs3nyapqgjev53ne