Is the set of all Heisenberg Matrices Connected? The set of all $3\times 3$ matrices of the form 
$$\begin{pmatrix}
1 & a & b\\
0 & 1 & c\\
0 & 0 & 1\\
\end{pmatrix}$$
forms a group surely, are they connected?

The obvious bijection from $\mathbf R^3$ (or is that $\mathbf C^3$? you didn't say) to this set of matrices, sending $(a,b,c)$ to the matrix you wrote, is a homeomorphism (continuous with continuous inverse). Therefore your question is equivalent to whether $\mathbf R^3$ (or $\mathbf C^3$) is connected, and this is true. The group structure plays no role at all for this.
