Collection of quasiextendable maps in functional spaces. (English) Zbl 0712.54012

There are numerous modifications of the notion of retract and related notions [comp. J. Klisowski, Colloq. Math. 46, 23-35 (1982; Zbl 0503.54022)]. In the present paper the author continues his studies of the notion of contraction, neighbourhood contraction, and related concepts. Let A be a non-empty subset of a metric compact space X and V a neighbourhood of A in X. A map r: \(X\to V\) is a contraction iff \(r(x)=x\) for every \(x\in A\); the set A is a contract of X iff for every V such a contraction exists; A is a neighbourhood contract of X if it is a contract of arbitrary neighbourhood of A in X. A space X is an absolute (neighbourhood) contract iff for every Y and every homeomorphism h of X onto a closed subset of Y, the set h(X) is a (neighbourhood) contract of Y. It is known that the class AS of absolute contracts coincides with FAR.
The author investigates the set F of quasiextendable maps of a compactum X into Y. Theorem 1.3 says that F is closed in \(Y^ X\). Theorem 1.4 says that \(Y^ X\) is AS if and only if Y is AS. Further, the author gives some characterizations of the classes AS and ANS.
Reviewer: M.Moszyńska


54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
54C15 Retraction
54C20 Extension of maps
54C35 Function spaces in general topology


Zbl 0503.54022