Let $O(n)$, $U(n)$, $SO(n)$ and $SU(n)$ be the orthogonal, unitary, special orthogonal and special unitary group respectively. What are the topology of these groups? (I'm just a beginner to algebraic topology, don't know much of differential topology). I am unable to find the answer to this question on the net, can someone tell me what the open sets of these topological spaces are?
My book just defines the sets and then goes on to prove that $SO(3)$ is homeomorphic to $RP^3$ without defining the topology of these groups.
All these groups are subgroups of the monoid $M = M(n,\mathbb R)$ of real $(n \times n)$-matrices (note that the monoidal structure on $M$ is given by matrix multiplication). But $M(n,\mathbb R)$ is a real vector space of dimension $n^2$ which can be identified via a linear isomorphism $\phi$ with $\mathbb R^{n^2}$. This gives us a topology on $M$ making $\phi$ a homeomorphism. It is easy to see that this topology does not depend on the choice of $\phi$. Your groups receive the subspace topology. Moreover, matrix multiplication in $M$ is easily seen to be continuous with respect to this topology. Finally, on the subgroup $GL(n,\mathbb R)$ of $M$ matrix inversion is easily seen to be continuous. Both properties transfer to your groups.