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A Structure Theorem for Topological Lattices

Published online by Cambridge University Press:  18 May 2009

L. W. Anderson  and
L. E. Ward Jr
L. E. Ward Jr
Affiliation:
University of Oregon, U.S. Naval Ordnanace Test Station
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In the study of connected partially ordered spaces a problem of fundamental interest is to determine sufficient conditions to ensure the existence of chains (i.e., simply ordered subsets) which are connected. Recently [5] R. J. Koch proved that, if X is a compact Hausdorff space with continuous partial order (i.e., the partial order has a closed graph), if L(x) = {y: y ≦ x} is connected for each x ∈ X, and if X has a zero (i.e., an element 0 such that 0 ≦ x for all x ∈ X), then each element of X lies in a connected chain containing zero. It is easy to find simple examples which show that this result is false if X is assumed only to be locally compact. However, if it is assumed that the partial order is that of a topological lattice then the existence of such chains can be shown by elementary methods. This solves a problem which was proposed in [3].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 5 , Issue 1 , January 1961 , pp. 1 - 3
Copyright
Copyright © Glasgow Mathematical Journal Trust 1961

References

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