Tree-like continua and cellularity

Abstract:

In this paper the equivalence of tree-like and cellular is proved for $1$-dimensional continua in ${E^n}$. More precisely, if $X$ is a tree-like continuum, then the collection of all embeddings $h:X \to {E^n},n \geqq 3$, such that $h[X]$ is cellular in ${E^n}$ is a dense ${G_\delta }$-subset of the collection of all maps from $X$ into ${E^n}$. Conversely, if $X$ is a $1$-dimensional cellular subset of ${E^n}$, then $X$ is a tree-like continuum.