Original scientific paper

https://doi.org/10.3336/gm.42.1.08

Closed embeddings into Lipscomb's universal space

Ivan Ivanšić ; FER, University of Zagreb, Unska 3, 10000 Zagreb, Croatia
Uroš Milutinović ; FNM, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia

Abstract

Let J(τ) be Lipscomb's one-dimensional space and Ln(τ) = {x J(τ)n+1 | at least one coordinate of x is irrational} J(τ)n+1 Lipscomb's n-dimensional universal space of weight τ ≥ אo. In this paper we prove that if X is a complete metrizable space and dim X ≤ n, w X ≤ τ, then there is a closed embedding of X into Ln(τ). Furthermore, any map f : X → J(τ)n+1 can be approximated arbitrarily close by a closed embedding ψ : X → Ln(τ). Also, relative and pointed versions are obtained. In the separable case an analogous result is obtained, in which the classic triangular Sierpinski curve (homeomorphic to J(3)) is used instead of J(אo).

Keywords

Covering dimension; embedding; closed embedding; universal space; generalized Sierpinski curve; Lipscomb's universal space; extension; complete metric space

Hrčak ID:

12886

URI

https://hrcak.srce.hr/12886

Publication date:

12.6.2007.

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