Open subset of upper triangular matrices Let $G$ be the group of all $2\times2$ non singular upper triangular matrices (with matrix multiplication) with entries in $\mathbb{R}$.View G as a topological subspace of $\mathbb{R^4}$.Let U be a subgroup of G which is an open subset in $G$.Prove that
$U$$\supseteq${$\begin{pmatrix}a& b\\0& d\end{pmatrix},\;\;$$a,d$ $\in$$\mathbb{R}_{>0}$, $b \in \mathbb{R}$}
 A: Since $U$ is a subgroup of $G$ it follows that 
$\begin{pmatrix}1& 0\\0& 1\end{pmatrix} \in U$.
Since the topology of $G$ is induced by the topology of $\mathbb R^4$ and $U$ is open in $G$ it follows that there exists $r > 0$ such that 
$\begin{pmatrix}1 + \delta & \mu \\0& 1 + \nu \end{pmatrix} \in U$ 
for all $\delta, \mu, \nu \in ]-r,r[$. Since
$\begin{pmatrix}1 + \delta & 0 \\0& 1 + \nu \end{pmatrix}^2  = \begin{pmatrix} (1 + \delta)^2 & 0 \\0& (1 + \nu)^2 \end{pmatrix}$
it follows that
$\begin{pmatrix}a & 0 \\0& d \end{pmatrix} \in U$
for all $a,d \in \mathbb R_{> 0}$, since for given $a,d \in \mathbb R_{> 0}$ there exists $k \in \mathbb N$ and $\delta, \nu \in ]-r,r[$ such that $a = (1 + \delta)^k$ and $b = (1 + \nu)^k$. Similarly it follows that 
$\begin{pmatrix} 1 & b \\0& 1 \end{pmatrix} \in U$ 
for all $b \in \mathbb R$ by solving $\begin{pmatrix}1 & \mu \\0& 1 \end{pmatrix}^k = \begin{pmatrix}1 & b \\0& 1 \end{pmatrix}$, for $\mu \in ]-r,r[$. 
Since $\begin{pmatrix}1 & b \\0& 1 \end{pmatrix}\begin{pmatrix}a & 0 \\0& d \end{pmatrix} = \begin{pmatrix}a & bd \\0&d \end{pmatrix}$ the claim follows.
