Consider the topological space $\mathbb{R}_S$ where $B=\{[a,b):a,b\in\mathbb{R}\}$ is a base for its topology. My textbook says that the set $D:=\{(x,x):x\in \mathbb{R}\}$ is a discrete subspace of $X=\mathbb{R}_S\times\mathbb{R}_S$. However I can't see how write each element of $D$ as an intersection of $D$ with an open set of $X$.
If I consider the set $D'=\{(x,-x):x\in \mathbb{R}\}$, it is easy to see that $D'$ is discrete so, it make me to think that there is a typo in my book.
Can someone help me to eliminate this doubt?
In any space $X \times X$, its diagonal is homomeomorphic to $X$. So the diagonal is a mistake.
In the Sorgenfrey plane $\mathbb{S} \times \mathbb{S}$, the antidiagonal $D'$ is indeed discrete as
$$D' \cap \left([x,x+1) \times [-x,-x+1)\right) = \{(x,-x)\}$$ for all $x$ and this is a relatively open subset of $D'$ as a subspace of $\mathbb{S} \times \mathbb{S}$ (the intersection of a product open subset with $D'$).