Continuous multi-utility representations of preorders

Normally preordered spaces and continuous multi-utilities

“…By our assumptions, we have that X \ A = We recall that a topological space (X, τ ) is perfectly normal if and only if it is a normal topological space and every closed subset of X is a G δ -set. The previous Proposition and Theorem 4.3 in Bosi and Herden [4] allow us to prove the following proposition. Proof.…”

“…The present paper can be viewed, in some sense, as a continuation of the paper [4] by Bosi and Herden. In that paper, the authors proved that a topological space (X, τ ) is normal iff every D-I-closed preorder on X gives (X, τ, ) the structure of a normally preordered space.…”

“…The next proposition follows from Theorem 4.3 and from the proof of Theorem 4.8 in Bosi and Herden [4]. Proposition 3.8.…”

“…Bosi and Herden [4,Theorem 4.3] proved that a space (X, τ ) is normal if and only if every D-I-closed preorder gives (X, τ, ) the structure of a normally preordered space. We now show that a similar result does not hold for regular spaces.…”

“…Let (X, τ ) be perfectly normal and be a D-I-closed preorder on X. From Bosi and Herden [4,Theorem 4.3 ], it follows that (X, τ, ) is normally preordered. Now, suppose that F is a decreasing open subset of X.…”