Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 21H / a / Solution

Write $v(x,y,z)$ for the displayed upper-unitriangular matrix. Matrix multiplication givesThus $V$ is the real Heisenberg group and $\Gamma$ is its Integer Heisenberg group. The subgroup $\Gamma$ is discrete. Choose an identity neighbourhood $U\subset V$ so small thatThen the translates $\gamma U$ are pairwise disjoint, and the quotient map restricts on each one to a homeomorphism onto the same open subset of $N$. Hence every point has an evenly covered neighbourhood andThis is the general quotient of a topological group by a discrete subgroup construction.