For every $x\in\mathbb R$ we have $\mathcal{B}(x):=\{[x,z): z>x\}$ as the sorgenfrey line.
First I want to show that there is topology $\tau$ defined on $\mathbb R$, i.e I have to show that $\mathcal{B}(x)$ satifies the properties of a neighborhood basis. I succeded in showing two parts, the last one $\forall V\in\mathcal{B}(x)\exists V_0\in\mathcal{B}(x)\forall z\in V_0\exists W\in\mathcal{B}(z):W\subseteq V$ is still remaining. Graphically its clear, but I do not know how to write it down formally.
Secondly I have a non-empty bounded interval I, such that $(a,b)\subseteq I\subseteq [a,b]$ for $a,b\in\mathbb R$ with $a<b$. I want to show that $I$ is open with repsect to the topolgy $\tau$ $<=> b\notin I$
My idea: I already shows that if $(X,\tau)$ is a topologocal space and for every $x\in X$ there is a neighborhood basis $\mathcal{B}(x)$, then for $I\subseteq X$: $I$ open$<=>\forall x\in I\exists V\in \mathcal{B}(x): V\subseteq I$
May you have an idea how to use this (I think it should work with this lemma) to prove the equivalence.
For that last property it suffices to show is that if $y\in V\in\mathscr{B}(x)$, then there is a $W\in\mathscr{B}(y)$ such that $W\subseteq V$. Since $V\in\mathscr{B}(x)$, $V=[x,z)$ for some $z>x$, and since $y\in V$, $y<z$; now let $W=[y,z)$ and verify that it has the required properties.
For the second part, if $b\notin I$, what is $\bigcup_{x\in I}[x,b)$? And if $b\in I$, is there any $V\in\mathscr{B}(b)$ such that $V\subseteq I$?