Vol. 15, No. 4, 1965

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On a conjecture of R. J. Koch

L. E. Ward

Vol. 15 (1965), No. 4, 1429–1433

R. J. Koch proved that if X is a compact, continuously partially ordered space and if W is an open subset of X which has no local minima, then each point of W is the supremum of an order arc which meets X W. More recently he extended this result to quasi ordered spaces in which the sets E(x) = {y : x y x} are assumed to be totally disconnected and W is a chain. He conjectured that the latter hypothesis is superfluous, and we show here that Koch’s conjecture is correct.

As a corollary it follows that if X is a compact, continuously quasi ordered space with zero (i.e., a unique minimal element), if each set E(x) is totally disconnected, and if each set L(x) = {y : y x} is connected, then X is arcwise connected.

Mathematical Subject Classification
Primary: 54.56
Received: 2 September 1964
Published: 1 December 1965
L. E. Ward