Our purpose in this article is first, following [8], to prove that if $\alpha $, $\beta $ are any points of the open unit disc $D(0;1)$ in the complex plane ${ \bf C}$ and $r$, $s$ are any positive real numbers such that ${ \overline{ D}}( \alpha ;r) \subseteq D(0;1)$ and ${ \overline{ D}}( \beta ;s) \subseteq D(0;1)$, then there exist $t \in (0,1)$ and a homeomorphism $h : { \overline{ D}}(0;1) \rightarrow { \overline{ D}}(0;1)$ such that ${ \overline{ D}}( \alpha ;r) \subseteq D(0;t)$, ${ \overline{ D}}( \beta ;s) \subseteq D(0;t)$, $h \left[ { \overline{ D}}( \alpha ;r) \right] = { \overline{ D}}( \beta ;s)$ and $h = id$ on ${ \overline{ D}}(0;1) \setminus D(0;t)$, and second, following [9], to prove that if $q \in { \bf N} \setminus \{ 0, 1 \} $ and ${ \bf B}({ \bf 0};1)$ is the open unit ball in ${ \bf R}^{ q}$, while for any $t>0$, we set $f^{ (t)}( { \bf x} ) = \frac{ t { \bf x} }{ 1 + (t-1) \Vert { \bf x} \Vert }$, whenever ${ \bf x} \in { \overline{ \bf B}}({ \bf 0};1)$, then $f^{ (t)} \rightarrow id$ in $C^{ 1} \left( { \overline{ \bf B}}({ \bf 0};1) , { \bf R}^{ q} \right) $ as $t \rightarrow 1^{ +}$.
Categories: math.GM