On homeomorphisms and \(C^{1}\) maps

• Published in: arXiv.org 2018-04
• Main Author: Sofronidis, Nikolaos E
• Format: Article
• Language: English
• Subjects: Real numbers
• ISSN: 2331-8422

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 \(t \rightarrow 1^{+}\).