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Some results on absolute extensors for ultranormal and ultraparacompact spaces. (English) Zbl 0857.54019
Let \({\mathcal C}\) be the class of ultranormal or ultraparacompact spaces. It is proved that a space is an absolute (neighbourhood) retract for \({\mathcal C}\) if and only if it is an absolute (neighbourhood) extensor for \({\mathcal C}\). Recall that \(X\) is ultranormal (resp., ultraparacompact) if \(\text{Ind}(X)=0\) (resp., \(X\) is paracompact with \(\text{Ind}(X)=0\)).
It is also proved that \(X\) is an AR(compact) implies \(X\) is an AE(ultranormal\(+T_1\)) (Theorem 3.4), and every Lindelöf AR(0-dim) is an AE(ultranormal) (Theorem 3.5).
Reviewer’s remark: It is well-known that every AR(compact) is an AE(normal), so author’s Theorem 3.4 follows from that fact. Also, it is not difficult to show that every completely regular AR(0-dim) is compact, which implies Theorem 3.5 (even without the assumption that \(X\) is Lindelöf).
MSC:
54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
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