A universal separable metric space based on the triangular Sierpiński curve

Let Σ(3) be the triangular Sierpiński curve. Call the vertices of the triangles obtained during the construction of Σ(3) (with the exception of the first triangle) the rational points of Σ(3), and all other points the irrational points of Σ(3). results of Lipscomb [Trans. Amer. Math. Soc. 211 (1975) 143–160] and techniques and results of Milutinović [Ph.D. thesis, 1993], [Glas. Mat. Ser. III 27 (47) (1992) 343–364], we prove that Ln(3)={x∈Σ(3)n+1: at least one coordinate of x is irrational} is a universal space for all separable metrizable spaces of dimension ⩽n.