I would like to show that the Circle group $SO(2)$ is a topological group under multiplication. The requirements are
$\alpha$) The group operation is continuous $m:G \times G \rightarrow G $
$\beta$) The inverse is continuous: $x \rightarrow x^{-1}$ is continuous
I'm aware that the group is invertible and if I would think of the $SO(2)$ group as a group of angles rather than a group of matrices, it seems obvious that the group is continuous. However, I would like to show that the group is continuous under multiplication for a set of matrices of the form $ \left( \begin{array}{rr} a & b \\ -b & a\end{array}\right)$.
How can one show continuity for a matrix?
You can write matrix multiplication as polynomials in the entries, and those are continuous. For the inverse, you normally also need the fact that $\rm det$ is continuous (and, well, nonzero on the thing you're looking at) but in this case perhaps you don't since it's $1$.