Concerning Koch's theorem on the existence of arcs

L. E. Ward

Vol. 15 (1965), No. 1, 347–355

DOI: 10.2140/pjm.1965.15.347

Abstract

A theorem of R. J. Koch asserts that if $X$ is a compact space endowed with a partial order $\Gamma$ such that

(i) $\Gamma$ is a closed subset of $X\times X$,

(ii) there exists $0\in X$ such that $\left(0,x\right)\in \Gamma$ for each $x\in X$, and

(iii) for each $x\in X$ the set $L\left(x\right)=\left\{y:y\leqq x\right\}$ is connected, then each point of $X$ lies in a connected chain containing $0$. In particular, $X$ is arcwise connected. This is a corollary of the theorem: if $X$ is a compact space and $\Gamma$ is a partial order satisfying (i), and if $W$ is an open subset of $X$ such that each neighborhood of each point $x$ of $W$ contains a point $y\ne x$ with $\left(y,x\right)\in \Gamma$, then each point of $W$ is the supremum of a connected chain which meets $X-W$. A new proof of these results is presented.

The first of these theorems is generalized in several ways. The compactness is relaxed to local compactness and the assumption that each closed chain has a zero. Moreover, the existence of a zero need not be assumed. If the set $E$ of minimal elements is closed, then $E$ is joined by connected chains to all other points of $X$. If the set function $L$ is continuous, then $E$ is necessarily closed.

Mathematical Subject Classification

Primary: 54.55

Milestones

Received: 13 February 1964

Published: 1 March 1965