A Note on Piercing a Disk

In this note we give a sufficient condition that a 2-cell or disk in E3 be pierced by a tame arc at each point of its interior. This condition is simply that each arc in the disk be tame. The method of proof leans heavily upon some results of Bing and Moise as well as upon a theorem on the union of tame disks by one of the present authors. Remark. Lemma 7 of [2] asserts that if two tame disks in E3 intersect in a simple arc on the boundary of each, then the union of the two disks is a tame disk. The proof of this result is an application of Theorem 5.1 of [4]. It is then a simple matter to see that if ZJ is a disk in E3 having an interior which is triangulated in a locally finite manner into tame 2-cells, then D is locally tame at each of its interior points. If in addition D has a tame boundary, then D is tame by either Theorem 5.13 of [5] or Theorem 5.1 of [4]. These observations will be used in our proof.