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
$XW$. 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.
