# A group is a bounded subset of another group

What does it mean when people say that a group is a bounded subset of another group?

I need decide whether $$SO_2(\mathbb C)$$ is a bounded subset of $$\mathbb C^{2\times 2}$$. What definition should I use? What is the "distance" function?

• $SO_2(\mathbb C)$ is certainly closed (as a set of solutions of continuous equations). Thus, it is bounded iff it is compact, for any metric compatible with the standard topology on $\mathbb C^{2\times 2}$. – lisyarus May 7 at 21:57
• @lisyarus What are the definitions of closedness and compactness of a group? What is the definition of that metric? – user398843 May 7 at 22:02

Take any norm on $$\mathbb C^{2\times2}$$ (for instance $$\left\lVert\left[\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right]\right\rVert=\sqrt{\lvert a\rvert^2+\lvert b\rvert^2+\lvert c\rvert^2+\lvert d\rvert^2}$$) and use the distance $$d(M,N)=\lVert M-N\rVert$$.