= Solution
Write $v(x,y,z)$ for the displayed upper-unitriangular matrix. Matrix multiplication gives
$$
v(x,y,z)v(x',y',z')=v(x+x',y+y'+xz',z+z').
$$
Thus $V$ 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 that
$$
UU^{-1}\cap\Gamma=\{1\}.
$$
Then 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 and
$$
\boxed{\ V\longrightarrow\Gamma\backslash V\text{ is a covering map}.\ }
$$
This is the general <quotient of a topological group by a discrete subgroup> construction.
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