prove that Every interval in the dictionary order topology on $R×R$ is the union of open sets in $R_d×R$

Question

how to prove that

Every interval in the dictionary order topology on $R×R$ is the union of open sets in $R_d×R$..where $R_d $ is the discrete topology

as i know that ${0}×R$ is open in the product topology $R_d×R$ as well in $R×R$

Answer 1

I preassume that you are talking about open intervals.

$\langle x,y\rangle\in(\langle a,b\rangle,\langle c,d\rangle)\iff\left[[x=a\wedge b<y]\vee[a<x<c]\vee[x=c\wedge y<d]\right]$

so that:$$(\langle a,b\rangle,\langle c,d\rangle)=\bigcup_{x\in[a,c]}\{x\}\times A_x$$where $A_a=(b,\infty)$, $A_x=\mathbb R$ if $a<x<c$ and $A_c=(-\infty,d)$.

Here $A_x$ is for every $x\in[a,c]$ an open subset of $\mathbb R$ equipped with its usual order topology, and $\{x\}$ is for every $x\in[a,c]$ an open subset of $\mathbb R$ equipped with discrete topology.

So $\{x\}\times A_x$ is open in $\mathbb R_d\times\mathbb R$ for every $x\in[a,c]$.