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\(P\)-embeddings, AR and ANR spaces. (English) Zbl 1030.54016

If \(X\) is a topological space, \(B\subset X\) is said to be \(P_0\)-embedded if 1) \(B\) is the zero set of a continuous real function on \(X\), and 2) every pseudometric \(\lambda:B\times B\to\mathbb{R}\) which is continuous (with the product topology) extends to a continuous pseudometric on \(X\). This paper characterizes metrizable absolute retracts or absolute neighborhood retracts by the extendability of maps from \(B\) to \(X\) where \(B\) is \(P_0\)-embedded. It also proves Morita’s generalization [K. Morita, Fundamenta Math. 88, 1-6 (1975; Zbl 0304.55009)] of Borsuk’s homotopy extension theorem.

MSC:

54E35 Metric spaces, metrizability
54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
54C20 Extension of maps
54C56 Shape theory in general topology
18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)

Citations:

Zbl 0304.55009
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