Extending real maps defined on a subset of a disk

Abstract

Let K and Q denote, respectively, the unit circular disk and the unit square disk in E2. Let A: K—>[0, l] and -ir: (?—>[0, l] be defined by A(x, y) = (x2+y2)112 and 7r(x, y) =x. Suppose Fis a subset of a disk D and / is a mapping of T into [O, l]; we are concerned here with conditions under which / can be extended to a mapping of D onto [0, l] which is topologically equivalent to A or to it. Recall that a mapping u: A—>B is topologically equivalent to a mapping w: A'—tB' if there exist homeomorphisms h: A—>A' and k: B—*B' such that ku=wh. It was shown in [2] that if F is a compact O-dimensional subset of Int D, then any mapping/ of T into the open interval (0, 1) has an extension to a mapping of D onto [O, l] which is topologically equivalent to 7T (and hence also an extension equivalent to A). However, the existence of a single arc in T, even though this arc is a subset of a point-inverse of /, may make the desired extension impossible. For example, if F is a subset of Int D consisting of a horizontal interval A together with points Xi, xi, X3, • • • such that the sequence {x2n} converges to a midpoint of A from above and {x2„+i} converges to a midpoint of A from below, then the mapping/ defined hyf(A) = 1/2, f(xn) = 1/2 + 1/3« cannot be extended to a mapping equivalent to jr. We therefore restrict attention in this paper to the case in which the common part of T and the interior of D is totally disconnected. First we show that if F is a compact subset of a disk D whose intersection with the interior of D is totally disconnected and T does not contain the boundary of D, then any mapping/ of F into [1/2, l] such that rnBd 7?=/_1(l) can be extended to a mapping of D onto [O, l] which is topologically equivalent to jr. Using this result, we show that if F is a compact subset of a disk D whose intersection with the interior of D is totally disconnected and F contains the boundary of D, then any mapping/ of F into [1/2, l] such that Bd D =f~l(l) can be extended to a mapping of D onto [O, 1 ] which is topologically equivalent to A. It follows as a special case of results in [l] that these theorems do not directly generalize to higher dimensions. If n'=3, l^k^n, and