## Closed connected sets which remain connected upon the removal of certain, connected subsets

### Volume 5 / 1924

Fundamenta Mathematicae 5 (1924), 3-10
DOI: 10.4064/fm-5-1-3-10

#### Abstract

The purpose of this paper is to prove: Theorem: Suppose M is a closed connected set containing more than one point such that if g is any connected subset of M, then M-g is connected. Under these conditions M is a simple closed curve. Theorem: If M is an unbounded closed connected set which remains connected upon the removal of any unbounded connected proper subset, then M is either an open curve, a ray of an open curve or a simple closed curve J plus OP, a ray of an open curve which has O and only O in common with J. Theorem: Suppose M is an unbounded closed connected set such that if g is any bounded connected subset of M, then M-g is connected. Then M is not a continuous curve.