Supplementary Exercises*: Topological Groups: Problem 1 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 $H$ denote a group that is also a topological space satisfying the $T_{1}$ axiom. Show that $H$ is a topological group if and only if the map of $H\times H$ into $H$ sending $x\times y$ into $x\cdot y^{-1}$ is continuous.

If $H$ is a topological group, then $f(x,y)=x\cdot y^{-1}$ is continuous as the composite of the following two continuous functions: $x\times y\rightarrow x\times y^{-1}$ (by Exercise 10 of §18) and $x\times y\rightarrow x\cdot y$ . If $f$ is continuous then it is continuous in both variables (Exercise 11 of §18), and, in particular, $f(1,y)$ and $f(x,f(1,y))$ ( $x\times y\rightarrow$ $x\times f(1,y)\rightarrow$ $f(x,f(1,y))$ ) are continuous.

Subpages

Supplementary Exercises*: Topological Groups: Problem 1 Solution

Supplementary Exercises*: Topological Groups: Problem 1 Solution