#### Document Type

Conference Proceeding

#### Language

eng

#### Format of Original

7 p.

#### Publication Date

8-1983

#### Publisher

American Mathematical Society

#### Source Publication

Proceedings of the American Mathematical Society

#### Source ISSN

0002-9939

#### Original Item ID

doi: 10.1090/S0002-9939-1983-0702301-6; Shelves: QA1 .A5215 Storage S

#### Abstract

We investigate the conjecture that the complement in the euclidean plane *E ^{2}* of a set

*F*of cardinality less than the continuum

*c*can be partitioned into simple closed curves if

*F*has a single point. The case in which

*F*is finite was settled in [1] where it was used to prove that, among the compact connected two-manifolds, only the torus and the Klein bottle can be so partitioned. Here we prove the conjecture in the case where

*F*either has finitely many isolated points or finitely many cluster points. Also we show there exists a self-dense totally disconnected set

*F*of cardinality

*c*and a partition of

*E*into "rectangular" simple closed curves.

^{2}\F
## Comments

Published version.

Proceedings of the American Mathematical Society, Vol. 88, No. 4 (August 1983): 691-697. DOI. © American Mathematical Society 1983. Used with permission.