An imbedding problem

Abstract:

If $H$ is an uncountable collection of pairwise disjoint continua in ${E^n}$, each homeomorphic to $M$, then there exists a sequence from $H$ converging homeomorphically to an element of $H$. In the present paper the authors show that if $\{ {M_i}\}$ is a sequence of continua in ${E^n}$ which converges homeomorphically to ${M_0}$ and such that for each $i,{M_i}$ and ${M_0}$ are disjoint and equivalently imbedded, then there exists an uncountable collection $H$ of pairwise disjoint continua in ${E^n}$, each homeomorphic to $M$. For $n = 2,\;3$, and $n \geqq 5$ it is shown that one cannot guarantee that the elements of $H$ have the same imbedding as ${M_0}$.