Locally convex topological lattices

Stralka, Albert R.

doi:10.1090/S0002-9947-1970-0264625-4

Locally convex topological lattices
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Trans. Amer. Math. Soc. 151 (1970), 629-640 Request permission

Abstract:

The main theorem of this paper is: Suppose that L is a topological lattice of finite breadth n. Then L can be embedded in a product of n compact chains if and only if L is locally convex and distributive. With this result it is then shown that the concepts of metrizability and separability are equivalent for locally convex, connected, distributive topological lattices of finite breadth.

References

Paul Alexandroff, Über die Metrisation der im Kleinen kompakten topologischen Räume, Math. Ann. 92 (1924), no. 3-4, 294–301 (German). MR 1512216, DOI 10.1007/BF01448011
Lee W. Anderson, On the distributivity and simple connectivity of plane topological lattices, Trans. Amer. Math. Soc. 91 (1959), 102–112. MR 102575, DOI 10.1090/S0002-9947-1959-0102575-X
Lee W. Anderson, One dimensional topological lattices, Proc. Amer. Math. Soc. 10 (1959), 715–720. MR 107678, DOI 10.1090/S0002-9939-1959-0107678-7
Lee W. Anderson, On the breadth and co-dimension of a topological lattice, Pacific J. Math. 9 (1959), 327–333. MR 105465
L. W. Anderson, The existence of continuous lattice homomorphisms, J. London Math. Soc. 37 (1962), 60–62. MR 133272, DOI 10.1112/jlms/s1-37.1.60
Kirby A. Baker and Albert R. Stralka, Compact, distributive lattices of finite breadth, Pacific J. Math. 34 (1970), 311–320. MR 282895
Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053
R. P. Dilworth, A decomposition theorem for partially ordered sets, Ann. of Math. (2) 51 (1950), 161–166. MR 32578, DOI 10.2307/1969503
Eldon Dyer and Allen Shields, Connectivity of topological lattices, Pacific J. Math. 9 (1959), 443–448. MR 105464
F. B. Jones, A theorem concerning locally peripherally separable spaces, Bull. Amer. Math. Soc. 41 (1935), no. 6, 437–439. MR 1563107, DOI 10.1090/S0002-9904-1935-06111-7
F. B. Jones, Concerning certain linear abstract spaces and simple continuous curves, Bull. Amer. Math. Soc. 45 (1939), 623–628. MR 268, DOI 10.1090/S0002-9904-1939-07054-7
J. D. Lawson, Lattices with no interval homomorphisms, Pacific J. Math. 32 (1970), 459–465. MR 256363

The relation of breadth and codimension in topological semilattices

S. Mrówka, Further results on $E$-compact spaces. I, Acta Math. 120 (1968), 161–185. MR 226576, DOI 10.1007/BF02394609
Leopoldo Nachbin, Topology and order, Van Nostrand Mathematical Studies, No. 4, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1965. Translated from the Portuguese by Lulu Bechtolsheim. MR 0219042
J. H. Roberts, A problem of Treybig concerning separable spaces, Duke Math. J. 28 (1961), 153–155. MR 131858
Prabir Roy, Separability of metric spaces, Trans. Amer. Math. Soc. 149 (1970), 19–43. MR 263020, DOI 10.1090/S0002-9947-1970-0263020-1
E. D. Shirley and A. R. Stralka, Homomorphisms on connected topological lattices, Duke Math. J. 38 (1971), 483–490. MR 279776
L. B. Treybig, Concerning certain locally peripherally separable spaces, Pacific J. Math. 10 (1960), 697–704. MR 116309
Alexander Doniphan Wallace, Two theorems on topological lattices, Pacific J. Math. 7 (1957), 1239–1241. MR 94305
Gordon Thomas Whyburn, Analytic Topology, American Mathematical Society Colloquium Publications, Vol. 28, American Mathematical Society, New York, 1942. MR 0007095