Dubrovnik VI – Geometric Topology September 30 – October 7, 2007
	ABSTRACTS	pdf-version of this abstract

Eva Trenklerová
P. J. Šafárik University, Košice, Slovakia

Basic embeddings in the plane

In our contribution we shall talk about a new, constructive proof of the characterization of compacta which are basically embedded in the plane.

A subset $K$ of the plane is said to be basically embedded, if for each $f\in C(K)$ there exist $g,h\in C(\mathbb R)$ such that $f(x,y)=g(x)+h(y)$ for each $(x,y)\in K$. According to Sternfeld and a reformulation of Skopenkov, a compact set $K\subset \mathbb R^2$ is basically embedded if and only if $K$ does not contain arrays of arbitrary length, where an array is a sequence of points $\{(x_i,y_i)\in \mathbb R^2\mid i\in I\}$, with $I=\{1,2,\ldots,n\}$ or $I=\mathbb N$, such that either $x_{2i-1}=x_{2i}$ and $y_{2i}=y_{2i+1}$ for all $i\in I$ or $y_{2i-1}=y_{2i}$ and $x_{2i}=x_{2i+1}$ for all $i\in I$ and no two consecutive points are equal.