A cardinal generalization of C∗ -embedding and its applications

As for extending real-valued continuous functions or continuous pseudometrics on a subspace to the whole space, notions of z -, C∗ -, C -, P - and Pγ -embeddings are known. As a cardinal generalization of z -embedding, Blair defined in 1985 the notion of zγ -embedding with γ≥ω , where zω -embedding coincides with z -embedding. On the other hand, since Pω -embedding equals C -embedding, Pγ -embedding can be also regarded as a cardinal generalization of C -embedding. Recently Ohta asked if a cardinal generalization of C∗ -embedding can be defined so that this property plus Uω -embedding is equal to Pγ -embedding, and it is itself equals C∗ -embedding in case γ=ω . In this paper, we give a cardinal generalization of C∗ -embedding, called (P∗)γ -embedding, and answer this problem. As a characterization of (P∗)γ -embedding, we show that (P∗)γ -embedding naturally admits its description by using continuous maps from a subspace into the hedgehog with γ spines. We also give a new extension-like property called weak zγ -embedding, with which z - (respectively C∗ -, C - or Uω -) embedding equals zγ - (respectively (P∗)γ -, Pγ - or Uγ -) embedding.