On the structure of completely useful topologies

Abstract

Let X be an arbitrary set. Then a topology t on X is completely useful if every upper semicontinuous linear preorder on X can be represented by an upper semicontinuous order preserving real-valued function. In this paper we characterize in ZFC (<span style="text-decoration: underline;">Z</span>ermelo-<span style="text-decoration: underline;">F</span>raenkel + Axiom of <span style="text-decoration: underline;">C</span>hoice) and ZFC+SH (ZFC + <span style="text-decoration: underline;">S</span>ouslin <span style="text-decoration: underline;">H</span>ypothesis) completely useful topologies on X. This means, in the terminology of mathematical utility theory, that we clarify the topological structure of any type of semicontinuous utility representation problem.

Semicontinuous Planar Total Preorders on Non-Separable Metric Spaces

“…Remark 3.5. In Corollary 4.5 in Bosi and Herden [9] it was already observed that a metric space is non-separable if and only if there exists an upper semicontinuous non-representable total preorder definable on such space. However the proof of that result uses the theorem by Estévez and Hervés [22] mentioned in the Introduction, which is based on long chains instead of planar chains.…”

“…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).…”

“…It can be proved that SRP always implies CRP, but the converse is not true in general (see e.g. Bosi and Herden [9]). …”

Order Embeddings with Irrational Codomain: Debreu Properties of Real Subsets

“…Also, the topology τ is said to satisfy the semicontinuous representability property (SRP) if every τ -upper (respectively τ -lower) semicontinuous total preorder defined on X admits a numerical representation by means of an upper (respectively lower) semicontinuous real-valued map. It is known that SRP implies CRP, but the converse is not true (See Proposition 4.4 in [7]). …”

Unified Representability of Total Preorders and Interval Orders through a Single Function: The Lattice Approach

“…With different nuances, the concept of a scale 1 (or similar notions, as R-separable systems) has already been used to get continuous or semicontinuous representations of total preorders (see e.g. [2,8,9,11,14,29,30]) and interval orders (see e.g. [6,12]).…”