Order-induced topological properties

Abstract

Each topology J on a set X may be associated with a preorder relation R& on X defined by (a, b) G R& iff every open set containing b contains a. Although the correspondence is many-to-one, there is always a least topology, ii(R), and a greatest topology, v(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 T p be an order-induced topological property with associated order property K p . We characterize minimal and maximal T p as follows: A topological space (X, 5~) is maximal T p iff 3~ = v{R^) and R* is minimal K p . With the imposition of a further condition on the class K p (satisfied by most properties under discussion), (X, 3~) is minimal T p iff SF = /X(JR^) and R°r is maximal K p . We apply these general theorems to a number of order-induced properties and conclude with an example to show that, for two particular properties, 3m ay be minimal T p even though R& is not maximal K p . Introduction.Correspondences between topologies and preorders on X similar to that assigning R^ to

Quasiorders, principal topologies, and partially ordered partitions

“…The result follows easily from the equivalence of these three statements: (1) N(x) C_ N(y), (2) y 6 cl{x}, (3) y <_ z. [3 Doyle [7] defines a tower space on n points to be any space homeomorphic to n with the chain topology {@, 1, 2, 3,..., _n_n}. More generally, we will say a chain topology, r on X is a tower topology on X if for every U 6 " there exists V 6 -and x 6 X \ V with V t3 {x} U. Doyle shows that a finite topological space X is a T0-space if and only if there is a continuous one-to-one function from X to a tower space.…”

Minimal TUD spaces

“…(3) The first case for Y X0 (x) and the fact that A is a set of minimal points in X 1 suffices. (4) The second case for Y X1−X0 (x) leaves the possibility that there is z ¡x 1 and…”

“…Given a topological space (X, T ), (X, T ) is -minimal Hausdorff if and only if it is Hausdorff and every open filterbase which has a unique adherent point is convergent to this point (see [5], [9], [10], [27], [31], [32], and [36]) -minimal T 1 if and only if T is the cofinite topology C on X -minimal regular if and only if it is regular and every regular filter-base which has a unique adherent point is convergent ([4], [8] [12], [19], [22], [26]) -minimal T D if and only if it is T D and nested ( [1], [12], [19], [22], [26] of X and some partition P of X such that P is simply associated with K and is associated with X \ K. ( [16]) -minimal T DD if and only if T = W K (P) ∨ (C ∩ I(K)) for some subset K of X and partition P of X such that P is simply associated with K and associated with X \ K ( [16]…”

On ultraconnected spaces

“…A topological space is ultraconnected if the intersection of any two nonempty closed sets is nonempty (Steen and In 1978 Andlma and Thron [2] defined a topological space (X,) to be upward directed if any two elements in (X,p()) have an upper bound, and it can easily be seen that the notion of upward directed and that of ultraconnected are equivalent.…”

“…In [2], it is proved that a topological space (X,z) is maximal upward directed iff (X,p(z)) is a partially ordered set of length I, with a greatest element and T V(p(T)).…”