TITLE: Given a matrix group, how to determine whether its connected , dense etc.
QUESTION [13 upvotes]: While going through Matrix groups , the problem I am facing is that how to determine different topological aspects of them, to be specific, Compactness is not a big problem due to Heine-Borel theorem , but dealing with other properties such as Connectedness and Density is not as easy. 
Firstly, I tried to think geometrically (like $SO(2) \cong S^1$ , $SU(2) \cong S^3$ , here $S^n$ denotes the group of the n-sphere in ${\Bbb R}^{n+1}$ , so $S^1$ denotes the circle group etc. ). But I am not able to do it for all the groups there on the table.
Next, for connectedness, I tried to do it with by considering continuous functions from that group to $\{1,-1\}$ , but still am facing difficulty.
And for density in particular, I have no clue.
Please help me by providing some strategies on how to learn and solve problems on Connectedness and Density of Matrix groups.

REPLY [2 votes]: As requested, here is an extended form of my comment. 

For connectedness of a compact Lie group $G$ acting transitively on a space $X$  (say, a manifold), you have a fiber bundle
$$
G_x\to G\to X, x\in X.
$$
Hence, if $X$ is connected and $G_x$ is connected, then $G$ is also connected. This can be viewed as a special case of the long exact sequence of homotopy groups of a fibration (see e.g. Hatcher's "Algebraic Topology"). 
For density, one needs much more information.