Useful topologies and separable systems

Abstract

Abstract. Let X be an arbitrary set. A topology t on X is said to be useful if every continuous linear preorder on X is representable by a continuous real{valued order preserving function.Continuous linear preorders on X are induced by certain families of open subsets of X that are called (linear) separable systems on X. Therefore, in a rst step useful topologies on X will be characterized by means of (linear) separable systems on X. Then, in a second step particular topologies on X are studied that do not allow the construction of (linear) separable systems on X that correspond to non{representable continuous linear preorders. In this way generalizations of the Eilenberg{Debreu theorems which state that second countable or separable and connected topologies on X are useful and of the theorem of Est evez and Herv es which states that a metrizable topology on X is useful, if and only if it is second countable can be proved.2000 AMS Classi cation: 54F05, 91B16, 06A05.

On a Possible Continuous Analogue of the Szpilrajn Theorem and its Strengthening by Dushnik and Miller

“…The particular selection of D implies, in addition, that the family {O i \ V} i∈I of open subsets of X is locally finite. With the help of the construction in the proof of Proposition 6.1 in Herden and Pallack [31] that is a generalization of an earlier construction of Estévez and Hervés [20] we, therefore, may construct some continuous total preorder on (X, t) that is not short. The following observation is made for later used (see Proposition 8.5):…”

“…With the help of the construction in the proof of Proposition 6.1 in Herden and Pallack [31] Proposition 8.6 implies that a first countable topological space (X, t) that has the Szpilrajn property and does not satisfy ccc allows the construction of a continuous total preorder on (X, t) that is not short. Hence, Lemma 8.1 allows us to conclude that the following corollary of Proposition 8.6 holds.…”

Continuous Utility Representation Theorems in Arbitrary Concrete Categories

“…the description in the second introductory section of Herden and Pallack [13] of a basis of t l and t r respectively with help of a basis of t) we may conclude that the inequalities β(X, t l ) ≤ β(X, t) and β(X, t r ) ≤ β(X, t) hold. t l ∪ t r is a sub-basis of t .…”

“…With help of this concept the proof of Proposition 6.1 in Herden and Pallack [13] implies the following generalization of the utility representation theorem of Estévez and Hervés.…”

Semicontinuous Planar Total Preorders on Non-Separable Metric Spaces

“…This is the continuous representability problem (or semicontinuous representability problem), see Herden and Pallack [25] and Bosi and Herden [9]. Thus, a topological space (X, τ ) is said to satisfy the continuous representability property CRP, (respectively, the semicontinuous representability property SRP) if every τ -continuous (respectively τ -semicontinuous) total preorder defined on X admits a representation by means of a real-valued order-preserving map U : (X, τ ) → (R, Euclidean topology) (i.e.,: x y ⇐⇒ U (x) ≤ U (y) (x, y ∈ X)), that is continuous (respectively, semicontinuous).…”