2-Dimension from the Topological Viewpoint

Barmak, Jonathan Ariel; Minian, Elías Gabriel

doi:10.1007/s11083-007-9057-1

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Finite Posets and Finite Spaces1

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2010

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“…Order preserving functions correspond to continuous maps and lower sets to open sets. For more details on finite spaces, we refer to reader to the foundational articles [12,21], P.May's notes [11] and the more recent articles [3][4][5]. See also [20] for the combinatorics of posets and [6,10] for the topology of posets.…”

Section: Finite Posets and Finite Spacesmentioning

confidence: 99%

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The Geometry of Relations

Minian

2010

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The classical way to study a finite poset (X, ≤) using topology is by means of the simplicial complex X of its nonempty chains. There is also an alternative approach, regarding X as a finite topological space. In this article we introduce new constructions for studying X topologically: inspired by a classical paper of Dowker (Ann Math 56:84-95, 1952), we define the simplicial complexes K X and L X associated to the relation ≤. In many cases these polyhedra have the same homotopy type as the order complex X . We give a complete characterization of the simplicial complexes that are the K or L-complexes of some finite poset and prove that K X and L X are topologically equivalent to the smaller complexes K X , L X induced by the relation <. More precisely, we prove that K X (resp. L X ) simplicially collapses to K X (resp. L X ). The paper concludes with a result that relates the K-complexes of two posets X, Y with closed relations R ⊂ X × Y.

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Section: Finite Posets and Finite Spacesmentioning

confidence: 99%

“…Finite spaces have been recently investigated in a series of joint papers with J. Barmak [3][4][5]. There is also a beautiful survey by P. May [11].…”

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The Geometry of Relations

Minian

2010

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2-Dimension from the Topological Viewpoint

Cited by 1 publication

References 9 publications

The Geometry of Relations

The Geometry of Relations

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