Geodesic
A Structure Theorem for Topological Lattices
Glasgow mathematical journal, Tome 5 (1961) no. 1, pp. 1-3

Voir la notice de l'article provenant de la source Cambridge University Press

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]. 
Jr, L. E. Ward. A Structure Theorem for Topological Lattices. Glasgow mathematical journal, Tome 5 (1961) no. 1, pp. 1-3. doi: 10.1017/S2040618500034213
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[1] 1.Anderson, L. W., On the distributivity and local connectivity of plane topological lattices, Trans. Amer. Math. Soc. 91 (1959), 102–112. Google Scholar

[2] 2.Anderson, L. W., One-dimensional topological lattices, Proc. Amer. Math. Soc. 10 (1959), 715–720. Google Scholar

[3] 3.Anderson, L. W., On the breadth and codimension of a topological lattice, Pacific J. Math. 9 (1959), 327–333. Google Scholar | DOI

[4] 4.Anderson, L. W. and Ward, L. E. Jr, One-dimensional topological semilattices. To appear in Illinois J. Math. Google Scholar

[5] 5.Koch, R. J., Arcs in partially ordered spaces, Pacific J. Math. 9 (1959), 723–728. Google Scholar | DOI

[6] 6.Ward, L. E. Jr, Partially ordered topological spaces, Proc. Amer. Math. Soc. 5 (1954), 144–161. Google Scholar

[7] 7.Wilder, R. L., Topology of manifolds (New York, 1949). Google Scholar | DOI

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