Supplementary Exercises*: Topological Groups: Problem 2 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

Show that the following are topological groups:

(a) $(\mathbb{Z},+)$

(b) $(\mathbb{R},+)$

(d) $(S^{1},\cdot)$ , where we take $S^{1}$ to be the space of all complex numbers $z$ for which $|z|=1$ .

(e) The general linear group $GL(n)$ , under the operation of matrix multiplication. ( $GL(n)$ is the set of all nonsingular $n$ by $n$ matrices, topologized by considering it as a subset of euclidean space of dimension $n^{2}$ in the obvious way.)

(a) (b) (c) The spaces are Hausdorff, the operations are continuous (Exercise 12 of §21, and for $\mathbb{Z}$ we don’t even need that).

(d) The space is Hausdorff (as a subspace of a Hausdorff space). Let $f(x,y)=x\cdot y^{-1}$ . If $U$ is open in $S^{1}$ , and $x\times y\in f^{-1}(U)$ , then $x\cdot y^{-1}\in U\subset S^{1}$ , and if $x=e^{i\phi}$ and $y=e^{i\theta}$ , then $z=x\cdot y^{-1}=e^{i(\phi-\theta)}$ . There is some $\epsilon>0$ such that $z\in A_{\phi-\theta,\epsilon}\subset U$ where $A_{\alpha,\delta}=\{e^{i\gamma}|\alpha-\delta<\gamma<\alpha+\delta\}$ , and for $x'\in A_{\phi,\epsilon/2}$ and $y'\in A_{\theta,\epsilon/2}$ , we have $f(x',y')\in A_{\phi-\theta,\epsilon}\subset U$ . Since $x\times y\in A_{\phi,\epsilon/2}\times A_{\theta,\epsilon/2}\subset f^{-1}(U)$ , and $A_{\phi,\epsilon/2}\times A_{\theta,\epsilon/2}$ is open in $S^{1}\times S^{1}$ , $f^{-1}(U)$ is open in $S^{1}\times S^{1}$ , and $f$ is continuous. By Exercise 1, $(S^{1},\cdot)$ is a topological group.

(e) It is Hausdorff (as a subspace of a Hausdorff space). The operation of multiplication can be represented as a product of compositions of addition and multiplication (using Exercise 10 of §18 and Exercise 12 of §21). This should help.

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Supplementary Exercises*: Topological Groups: Problem 2 Solution

Supplementary Exercises*: Topological Groups: Problem 2 Solution