Geodesic
Maximal point spaces of posets with relative lower topology
Filomat, Tome 35 (2021) no. 8, p. 2645

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DOI

In domain theory, by a poset model of a T 1 topological space X we usually mean a poset P such that the subspace Max(P) of the Scott space of P consisting of all maximal points is homeomorphic to X. The poset models of T 1 spaces have been extensively studied by many authors. In this paper we investigate another type of poset models: lower topology models. The lower topology ω(P) on a poset P is one of the fundamental intrinsic topologies on the poset, which is generated by the sets of the form P ↑x, x ∈ P. A lower topology poset model (poset LT-model) of a topological space X is a poset P such that the space Max ω (P) of maximal points of P equipped with the relative lower topology is homeomorphic to X. The studies of such new models reveal more links between general T 1 spaces and order structures. The main results proved in this paper include (i) a T 1 space is compact if and only if it has a bounded complete algebraic dcpo LT-model; (ii) a T 1 space is second-countable if and only if it has an ω-algebraic poset LT-model; (iii) every T 1 space has an algebraic dcpo LT-model; (iv) the category of all T 1 space is equivalent to a category of bounded complete posets. We will also prove some new results on the lower topology of different types of posets
DOI : 10.2298/FIL2108645S
Classification : 54A10, 06B35, 54A20
Keywords: Lower topology, Scott topology, maximal point space, lower topology poset model, poset model 
Chong Shen; Xiaoyong Xi; Dongsheng Zhao. Maximal point spaces of posets with relative lower topology. Filomat, Tome 35 (2021) no. 8, p. 2645 . doi: 10.2298/FIL2108645S
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     author = {Chong Shen and Xiaoyong Xi and Dongsheng Zhao},
     title = {Maximal point spaces of posets with relative lower topology},
     journal = {Filomat},
     pages = {2645 },
     year = {2021},
     volume = {35},
     number = {8},
     doi = {10.2298/FIL2108645S},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2108645S/}
}
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