I have to show that

$\displaystyle \mathcal{T} = \left\{ (a, b] : a<b, a,b \in \mathbb{R}\right\}$

is a topology. I know that this is not a topology, because it forms the basis for a topology on $\displaystyle \mathbb{R}$. But why is this the case? I am thinking of the case, for instance, $\displaystyle (1,3] \cup (5,7]$. I don't think this is in $\displaystyle \mathcal{T}$, or am I incorrect?

Thanks in advance,

HTale.