Construction of functions with given cluster sets
Tom 152 / 2018
Streszczenie
We continue our study of functions on the boundary of their domain and obtain some results on cluster sets of functions between topological spaces. For a compact space $\overline {Y}$ the cluster set of a function $f:D\to \overline {Y}$ at a point $x\in \overline {D}$ is the set $\overline {f}(x)=\bigcap \{\overline {f(U\cap D)}: U$ is a neighborhood of $x\}$ and it equals the $x$-section of the closure of the graph of $f$. We prove that for a metrizable topological space $X$, a dense subspace $Y$ of a metrizable compact space $\overline {Y}$, a closed nowhere dense subset $L$ of $X$, an upper continuous compact-valued multifunction ${\varPhi :L\multimap \overline {Y}}$ and a set $D\subseteq X\setminus L$ such that $L\subseteq \overline {D}$, there exists a function $f:D\to Y$ such that the cluster set $\overline {f} (x)$ is equal to $\varPhi (x)$ for any $x\in L$.
