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Keywords:
sober topological space; Scott topology; weak topology
sober topological space; Scott topology; weak topology
Summary:
We give some necessary and sufficient conditions for the Scott topology on a complete lattice to be sober, and a sufficient condition for the weak topology on a poset to be sober. These generalize the corresponding results in [1], [2] and [4].
References:
[1] Isbell J.R.: Function spaces and adjoints. Math. Scand. 36 (1975), 317-339. MR 0405340 | Zbl 0309.54016
[2] Hoffmann R.-E.: Essentially complete $T_0$ spaces. Manuscripta Math. 27 (1979), 401-432. MR 0534803 | Zbl 0418.54020
[3] Isbell J.R.: Completion of a construction of Johnstone. Proc. Amer. Math. Soc. 85 (1982), 333-334. MR 0656096 | Zbl 0492.06006
[4] Gierz G., Hoffmann K.H., Keimel K., Lawson J.D., Mislove M., Scott D.S.: A Compendium of Continuous Lattices. Springer-Verlag, 1980. MR 0674650
[5] Johnstone P.T.: Scott is not always sober. Lecture Notes in Math. 871, Springer-Verlag, 1981, pp.282-283. Zbl 0469.06002
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