Uro¹ Milutinoviæ*
University of Maribor, Slovenia
Closed embeddings into Lipscomb's universal space
$\mathcal J(\tau)$ be Lipscomb's one-dimensional
space and $L_n(\tau)$ $=\{x\in\mathcal J(\tau)^{n+1}\, |$
{at least one coordinate of $x$ is irrational}$\}\subseteq{\mathcal J(\tau)}^{n+1}$ Lipscomb's $n$-dimensional universal
space of weight $\tau\geq\aleph_0$. We prove that if $X$ is a complete
metrizable space and $\dim X\leq n$, $w X\leq\tau$,
then there is a closed embedding of $X$ into $L_n(\tau)$. Furthermore, any map
$f\colon X\to\mathcal J(\tau)^{n+1}$ can be approximated arbitrarily close by a closed embedding $\psi\colon X\to L_n(\tau)$.
Also, relative and pointed versions are obtained. In the separable case
an analogous result is obtained, in which the classic triangular
Sierpi\'nski curve (homeomorphic to $\mathcal J(3)$) is used instead
of $\mathcal J(\aleph_0)$.
* This is a joint work with Ivan Ivan¹iæ.
