Functional extenders and set-valued retractions

Abstract

a b s t r a c tWe describe the supports of a class of real-valued maps on C * (X) introduced by Radul (2009) [6]. Using this description, a characterization of compact-valued retracts of a given space in terms of functional extenders is obtained. For example, if X ⊂ Y , then there exists a continuous compact-valued retraction from Y onto X if and only if there exists a normed weakly additive extender u: C * (X) → C * (Y ) with compact supports preserving min (resp., max) and weakly preserving max (resp., min). Similar characterizations are obtained for upper (resp., lower) semi-continuous compact-valued retractions. These results provide characterizations of (not necessarily compact) absolute extensors for zero-dimensional spaces, as well as absolute extensors for one-dimensional spaces, involving non-linear functional extenders.Crown

Diffeomorphism groups of non-compact manifolds endowed with the WhitneyC∞-topology

“…Combining Theorem 3.9 with Shchepin's Theorem 3.6, we get: Since each monadic normal functor admit a tensor product (see [23, §3.4]), this corollary implies: Applying Corollary 3.11 to the functor of idempotent measures I (which monadic and normal [26]), we get the following corollary answering a problem posed in [1].…”

“…For a subspace A of X by A (1) we denote the set of non-isolated points of A and for every ordinal α define the α-th derived set X (α) of X letting X (0) = X, X (1) be the set of non-isolated points of X and (1) for α > 1. For a scattered space X the transfinite sequence (X α ) α is strictly decreasing, so X (α) = ∅ for some α.…”

“…Theorem 1.1 shows that Dugundji compact spaces are tightly connected with the functor of probability measures P (this was observed and widely exploited byŠčepin in [22]). The relations of the class of Dugundji spaces to some other functors was studied by Alkinson and Valov [1], [25].…”

F-Dugundji spaces, F-Milutin spaces and absolute F-valued retracts

“…The equivalences (2) ⇔ (3) and (1) ⇔ (4) follow from Theorem 2.3 while (3) ⇒ (4) is trivial. To prove that (4) ⇒ (3), assume that for some embedding X ⊂ K into a Tychonoff cube K = [0, 1] κ there exists a continuous map s :…”

“…For a subspace A of X by A (1) we denote the set of non-isolated points of A and for every ordinal α define the α-th derived set X (α) of X letting X (0) = X, X (1) be the set of non-isolated points of X and…”