Sorgenfrey topology

Question

$B$ is the base of the Sorgenfrey topology ($\mathcal{T}_{S}$ ), being $ \mathcal{T}_{u}$ the usual topology.

Answer 1

If you show that $[5,\infty)$ is open in $\mathcal{T}_S$, then your set will be open in any product space $X\times Y$ where $X=(\mathbb{R},\mathcal{T}_S)$, because $U\times Y$ is open in the box or product topology when $U\in \mathcal{T}_S$.

Notice on the other hand that $[5,\infty)=\cup_{n=1}^\infty[5,5+n)$, and hence open.