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On a conjecture of R.
J. Koch
L. E. Ward |
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Vol. 15 (1965), No. 4, 1429–1433
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Abstract |
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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
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Milestones
Received: 2 September 1964
Published: 1 December 1965
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Authors |
| L. E. Ward | |
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