I was going through the PhD thesis of Daniel Nagaj. In the last paragraph on page 30, there is this following sentence.
A universal gate set must be dense in the group $SU(n)\ldots$
My question is what does 'dense' mean when it is about a special unitary group?
The group $SU(n)$ of unitary $n\times n$ matrices with determinant $1$ can be viewed as a subset of $\Bbb{C}^{n^2}$, in the way that the set of all $n\times n$ matrices can be viewed as all of $\Bbb{C}^{n^2}$. Since $\Bbb{C}$ has a topology, $SU(n)$ inherits this topology when we interpret $SU(n)\subseteq\Bbb{C}^{n^2}$, so it makes sense to talk about a subset of $SU(n)$ being dense in $SU(n)$: it means that when thought of as a subset $S\subseteq\Bbb{C}^{n^2}$, $\overline{S} = SU(n)$ - which is exactly what it means to be dense normally. We simply need to interpret the first entry of a matrix $M\in SU(n)$ as the first coordinate of a vector in $\Bbb{C}^{n^2}$, the entry in the first row and the second column as the second entry of a vector in $\Bbb{C}^{n^2}$ (with the standard basis, of course), and so on, to be able to talk about a topology on $SU(n)$, whence it becomes natural to talk about denseness, openness, closedness, or any topological property of $SU(n)$ or a subset thereof.