Concerning the Dugundji extension property

Extending pointwise bounded equicontinuous collections of functions

“…$PR\infty F$ . The proof is based on Stares [18]. Let $X$ be a decreasing $(G)$ space, $A$ a non-empty closed subspace of $X$ and $Y$ an LCTV-space.…”

“…pointwise totally bounded) equicontinuous subset $\{f_{\alpha} : \alpha\in\Omega\}$ of $C(A, Y)$ , the collection { $\Phi(f_{\alpha})$ : $\alpha\in\Omega\}$ is pointwise bounded (resp. pointwise totally bounded) equicontinuous, where $\Phi$ : $C(A, Y)\rightarrow C(X, Y)$ is Dugundji's extender constructed by Stares in [18].…”

Continuous extension of functions from countable sets

“…The existence and properties (linearity, continuity with respect different topologies, etc.) of such extender ϕ : C(A) → C(X) for various classes of spaces X were investigated by many mathmeticians (see, for instance, [3], [2], [10], [11], [12], [5], [14] and literature given there). In particular, the existence of a linear continuous extender ϕ : C p (A) → C p (X) for every closed subset A of a stratifiable space X was obtained in [2,Theorem 4.3] and the existence of such extender for every closed subset A of locally compact generalized ordered space X was proved in [5, Corollary 1]).…”

Monotone extenders for bounded c-valued functions

“…Dugundji's extension theorem [6] states that for a metric space X, a closed subset A of X and a locally convex topological vector space Y , there exists a linear conv-extender u : C(A, Y ) → C(X, Y ); this is an improvement of an earlier result by K. Borsuk [4] that for a closed separable subset of a metric space X, there exists a norm-one linear extender u : C ∞ (A) → C ∞ (X). Now it is known that Dugundji's extension theorem holds in some classes of generalized metric spaces X (C. J. R. Borges [3], I. S. Stares [11]), but does not hold for all GO-spaces X. Indeed, for the Michael line R Q , R. W. Heath and D. J. Lutzer [9] show that there exists no linear conv-extender u : C(Q) → C(R Q ); E. K. van Douwen [5] extends it by showing that there is no monotone extender u : C(Q) → C(R Q ) (see also I. S. Stares and J. E. Vaughan [12]).…”