Dugundji Extension Theorem Research Articles

A non-archimedean Dugundji extension theorem

We prove a non-archimedean Dugundji extension theorem for the spaces C*(X, C* (X, K) of continuous bounded functions on an ultranormal space X with values in a non-archimedean non-trivially valued complete field K. Assuming that K is discretely valued and Y is a closed subspace of X we show that there exists an isometric linear extender T: C* (Y, K) → K* (X, K) if X is collectionwise normal or Y is Lindelöf or K is separable. We provide also a self contained proof of the known fact that any metrizable compact subspace Y of an ultraregular space X is a retract of X.

ON A THEOREM OF ARVANITAKIS

Arvanitakis established recently a theorem which is a common generalization of Michael's convex selection theorem and Dugundji's extension theorem. In this note we provide a short proof of a more general version of Arvanitakis' result.

Dugundji’s theorem for cone metric spaces

In this work, we will prove the Dugundji extension theorem for the cone metric space. It is heavily reliant on the paracompactness of the cone topology that is proved by Ayse Sönmez in the paper Sönmez (2010) [11].

Some results on the extension of single and multivalued maps

Necessary and sufficient conditions for single-valued extensions of multivalued maps are discussed. Moreover, a quantitative version of a generalization of Dugundji's extension theorem for multivalued maps is obtained. Finally, the extension problem for compact maps is studied. Many of the results are new even for single-valued maps.

π-embeddings and Dugundji extension theorems for generalized ordered spaces

We study generalized ordered spaces in which every closed subspace is π-embedded and which satisfy the Dugundji Extension Theorem. We prove: Let X be a perfectly normal generalized ordered space in which the set E( X) = { x ϵ X: (→, x] or [ x,→) is open in X} is σ-discrete in X. Then every closed subspace of X is π-embedded. Furthermore, for every closed subspace A of X and for any locally convex linear topological space Z there is a linear transformation u : C( A, Z) → C( X, Z) such that for each f ϵ C( A, Z), u( f) is an extension of f and the range of u( f) is contained in the closed convex hull of the range of f. This is a partial answer to a question asked by Heath and Lutzer (1974).

Continuous Selections for Continuous Set-Valued Mappings and Finite-Dimensional Sets

The paper is devoted to selection theorems for ‘continuous’ set-valued mappings which are not necessarily convex-valued on finite-dimensional subsets of the domain. As a result, we get natural generalizations of the famous Kuratowski–Dugundji extension theorem. The technique, developed in the paper, applies to l.s.c. mappings with paracompact, or similar, domain as well. Also, it leads us to a new construction of set-valued continuous selections that involves dimension-type arguments.

Concerning the Dugundji extension property

From Borges' proof that stratifiable spaces have the Dugundji extension property we abstract a normality type condition and hence show that the Dugundji extension theorem is valid for an even bigger class of spaces. Borges claimed that linearly stratifiable spaces also have the Dugundji extension property. By considering cardinal generalisations of our normality condition we show that the proof of this is false.

The Dugundji extension theorem and extension degree

The Dugundji extension theorem and extension degree

Another generalization of the Dugundji extension theorem

Another generalization of the Dugundji extension theorem

On the random Dugundji extension theorem

Soit (Ω, #7B-B) un espace mesurable, X un espace metrique, A un sous-ensemble clos non vide de X, et Y un espace norme separable. Soit f: Ω×A→Y de Caratheodory. Alors il existe une fonction de Caratheodory f~: Ω×X→Y telle que: 1) f~ restreinte a Ω×A est egale a f, et 2) f~(ω,X)⊂cof(ω,A) pour chaque ω∈Ω

Some Extensions of the Tietze-Urysohn Theorem

In this note we examine certain sharpenings of the Tietze-Urysohn extension theorem. We prove, for example, that if A is a closed subspace of a normal space X then there is a continuous extender v,:C*(A) -C*(X), where C*(A) and C*(X) are the Banach spaces of continuous, bounded, real-valued functions on A and X, respectively. However, well-known examples show that we cannot arrange that q is both continuous and linear, or that q is an isometry. Introduction. The classical Tietze-Urysohn theorem guarantees that every continuous, bounded, real-valued function f defined on a closed subspace A of a normal space X can be extended to a continuous, real-valued function f defined on all of X and having the same bounds as f. If we let C*(A) and C*(X) denote the real vector spaces of continuous, bounded, real-valued functions on A and X respectively, then the Axiom of Choice yields the following slight reformulation of the classical theorem. A. THEOREM (Tietze-Urysohn). If A is a closed subspace of the normal space X then there is a function r: C*(A ) C*(X) such that for every f E C*(A), rl(f) extends f and has the same bounds as f Any function q:C*(A)-+ C*(X) having the property that 71(f) extends f whenever f E C*(A) will be called an extender from C*(A) to C*(X). It is natural to wonder whether, if one selects the extension 71(f) of f more carefully, one can obtain an extender r1 which respects the natural linear and/or topological structures of the function spaces in Theorem A. For metric spaces, an affirmative answer is provided by the Dugundji Extension Theorem. B. THEOREM [D]. Suppose A is a closed subset of a metrizable space X. Then there is a function 77: C*(A) -* C*(X) such that: (1) for each f E C*(A), r(f) extends f and has the same bounds as f; (2) the function 77 is linear.(2' Theorem B is a model for all of the extension theorems considered in this paper so we shall pause to make a few comments about its conclusions and its proof. In introducing the theorem we said that it shows that 7q can be constructed in such a way that 7q respects the natural topological structures of the function spaces. Let us now introduce one such structure, the sup-norm topology. For a function f E C*(A) the sup-norm of f is defined by lIf || = sup {if(x){:x E A}; the norm of a function F E C*(X) is analogously defined. When equipped with these norms, C*(A) and C*(X) are Banach spaces and the two conclusions of Theorem B force 77 to be an isometry. That may be seen as follows. Let f, g E C*(A). Then 1j 71(f) 71(g) jj = 1177 (f g) I since 77 is linear, and because 7q(f g) has the same bounds as f g, 1j 71 (f g) jj = jj f g 1j. Thus 7q is a linear, isometric extender. The proof of Theorem B is too technical to include here but its central idea is easily described, at least in principle. One does not extend the functions in C*(A) one at a time as in the proof of Theorem A; instead one defines an extension process which acts simultaneously (and linearly) on all members of C*(A). Dugundji's extension process has been improved in recent years so that it now 1. This paper was completed while the third author was visiting the University of Pittsburgh as a Mellon Postdoctoral Fellow. 2. Dugundji actually proved that a linear q can be found such that for each f E C*(A), v(f) extends f and the range of q (f) is contained in the convex hull of the range of f. Furthermore, if one considers bounded, continuous, complex-valued functions, then one can obtain complex-linear extenders with the property that the range of v1(f) is contained in the convex hull of the range of f.

On a generalization of Dugundji extension theorem

On a generalization of Dugundji extension theorem

A note on the Dugundji extension theorem

We prove that if A A is a closed, metrizable, G δ {G_\delta } -subspace of a collectionwise normal space X X then there is a linear transformation e : C ( A ) → C ( X ) e:C(A) \to C(X) such that for each g ∈ C ( A ) , e ( g ) g\in C(A),e(g) extends g g and the range of e ( g ) e(g) is contained in the closed convex hull of the range of g g .