A General Separation Theorem For Various Structures
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A GENERAL SEPARATION THEOREM FOR VARIOUS STRUCTURES F. JORDAN1,∗ , I. KALANTARI2 and H. PAJOOHESH3 1
Department of Mathematics and Computer Science, Queensborough Community College, Bayside, NY 11364, USA e-mail: [email protected] 2
3
Department of Mathematics, Western Illinois University, Macomb, Illinois 61455, USA
Department of Mathematics, Medgar Evers College, CUNY, Brooklyn, NY 11225, USA e-mail: [email protected] (Received October 27, 2019; revised March 31, 2020; accepted April 2, 2020)
Abstract. There are several important separation theorems in various areas; for example, theorems of Gordan, Stone, Mazur, Hahn–Banach, etc. In this paper, we give a general treatment, in ZFC, of separation results with several examples in old and new settings. In order to achieve some uniformity of the treatment, we define the notion of a solid operator that leads to the notion of separation. We also characterize those topological spaces for which the closure of a set is a solid operator. Further, we prove a separation theorem for solid operators which we will use to characterize those graphic matroids whose span is a solid operator. We also use the separation theorem to prove a theorem of P´ ales.
1. Introduction The closure of a set in a topological space, the convex hull of a set, and the span of a set in a vector space, for example, are in some ways similar. Specifically, under certain conditions, disjoint sets in those settings are known to be separable by complementary sets exhausting the whole space, a phenomenon used in significant settings and applications. Intrigued by the parallel, we pursue in this paper a general formulation of an operation and prove a result on ‘separation’ that is applicable to the cases for convex closure, topological closure, span, etc. It turns out that with small adjustments, our notion of ‘separation’ applies in other settings and examples such as matroids, graphic matroids and various topologies (see Sections 5–9) and leads to interesting results, as well. ∗ Corresponding
author. Key words and phrases: separation, solid operator, closure operator, convexity, topological closure, vector space, matroid. Mathematics Subject Classification: 06A15, 05B35. c 2020 0236-5294/$ 20.00 © 0 Akad´ emiai Kiad´ o, Budapest 0236-5294/$20.00 Akade ´miai Kiado ´, Budapest, Hungary
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F. F. JORDAN, JORDAN, I. I. KALANTARI KALANTARI and and H. H. PAJOOHESH PAJOOHESH
Our motivation is the existence of several ‘separation theorems,’ such as Gordan’s [3], Stone’s [7], Mazur’s [2] (the geometric form of Hahn–Banach’s [1] as well as some more well-known theorems of Banach space theory are treated in [2]). All these theorems have a form that says if you have two disjoint bodies that are closed under relevant operations, then you can find a ‘separator’ that is some large determinator of the space such that each of the given two bodies falls in ‘one side’ of the ‘separator’ (so they are on opposite sides). As separation and extension of linear forms are keenly related, there are sever
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