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.
