Vol. 75, No. 2, 1978

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Order-induced topological properties

Susan Jane Zimmerman Andima and W. J. Thron

Vol. 75 (1978), No. 2, 297–318
Abstract

Each topology 𝒯 on a set X may be associated with a preorder relation R𝒯 on X defined by a,b⟩∈ R𝒯 iff every open set containing b contains a. Although the correspondence is many-to-one, there is always a least topology, μ(R), and a greatest topology, ν(R), having a given preorder R. This leads to a natural correspondence between order properties and some topological properties and to the concept of an order-induced topological property. We show that a number of familiar topological properties (mostly lower separation axioms) are order-induced and also consider some new properties suggested by order properties. Let Tp be an order-induced topological property with associated order property Kp. We characterize minimal and maximal Tp as follows: A topological space (X,𝒯 ) is maximal Tp iff 𝒯 = ν(R𝒯) and R𝒯 is minimal Kp. With the imposition of a further condition on the class Kp (satisfied by most properties under discussion), (X,𝒯 ) is minimal Tp iff 𝒯 = μ(R𝒯) and R𝒯 is maximal Kp. We apply these general theorems to a number of order-induced properties and conclude with an example to show that, for two particular properties, 𝒯 may be minimal Tp even though R𝒯 is not maximal Kp.

Mathematical Subject Classification 2000
Primary: 54F99
Secondary: 54F05
Milestones
Received: 7 January 1976
Published: 1 April 1978
Authors
Susan Jane Zimmerman Andima
W. J. Thron