Ivan Ivanšić

A universal separable metric space based on the triangular Sierpiński curve

Let Σ(3) be the triangular Sierpinski 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). Using results of Lipscomb [Trans. Amer. Math. Soc. 211 (1975) 143–160] and techniques and results of Milutinovic [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.

Borsuk's index and pointed movability for projective movable continua

Abstract It is shown that each projective movable continuum X is shape dominated by a regularly movable continuum of the same dimension. This has two consequences. First, if the dimension of X is ≤k , k≠2 , then X is shape dominated by a continuum in R 2k . This answers affirmatively a special case of a question raised by Borsuk, at least as far back as 1975, in all dimensions except dim =2 . Second, it implies that such a continuum is pointed movable, again giving an affirmative answer, in a special case, to the old question in shape theory of whether movable continua are always pointed movable.