A universal separable metric space based on the triangular Sierpiński curve
The authors construct, for each \(n\), a space \(\Sigma(3)^{n+1}\) out of copies of the triangular Sierpiński curve \(\Sigma(3)\); co-author Milutinović has shown in earlier work that \(\Sigma(3)\) is homeomorphic to a construction of S. L. Lipscomb. They go on to define \(L_n(3)= \{x\in\Sigma(3)^{n+1}\): at least one coordinate of \(x\) is irrational\}, where an irrational point of \(\Sigma(3)\) is one that is not a vertex of one of the triangles formed in the construction of \(\Sigma(3)\). The authors then present a proof that this space \(L_n(3)\) is a universal space for all separable \(n\)-dimensional metric spaces. They go on to announce that the same methods will obtain the following Theorem (the details will appear elsewhere): Let \(X\) be any separable \(n\)-dimensional metric space, and \(f: X\to \Sigma(3)^{n+1}\) any continuous mapping. Then for any \(\varepsilon> 0\) there is an embedding \(\psi: X\to L_n(3)\subseteq \Sigma(3)^{n+1}\) such that \(d(f,\psi)\leq \varepsilon\).
separable \(n\)-dimensional metric space
scientific article; zbMATH DE number 1761006
scientific article