Supplementary Exercises*: Topological Groups: Problem 4 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Let $\alpha$ be an element of $G$ . Show that the maps $f_{\alpha},g_{\alpha}:G\rightarrow G$ defined by $\begin{eqnarray*} f_{\alpha}(x)=\alpha\cdot x & \mbox{ and } & g_{\alpha}(x)=x\cdot\alpha \end{eqnarray*}$ are homeomorphisms of $G$ . Conclude that $G$ is a homogeneous space. (This means that for every pair $x$ , $y$ of points of $G$ , there exists a homeomorphism of $G$ onto itself that carries $x$ to $y$ .)

Note that $f_{\alpha}\circ f_{\alpha^{-1}}$ , $f_{\alpha^{-1}}\circ f_{\alpha}$ , $g_{\alpha}\circ g_{\alpha^{-1}}$ and $g_{\alpha^{-1}}\circ g_{\alpha}$ are the identity maps, so, $f_{\alpha}$ and $g_{\alpha}$ (and their inverse functions) are bijective and continuous (Exercise 11 of §18). For every $x$ and $y$ , $f_{y\cdot x^{-1}}(x)=y$ , or $g_{x^{-1}\cdot y}(x)=y$ .

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Supplementary Exercises*: Topological Groups: Problem 4 Solution

Supplementary Exercises*: Topological Groups: Problem 4 Solution