# Relation induced topology

For an ordered space $$X$$ there is the term of ordered topology generated by sets of the form:

$$l(x)=\{ y\in X: y and $$r(x)=\{ y\in X: y>x\}$$

I was wondering if someone had encountered somewhere and can give a reference of reading materials on an a topology induced by relation. More precisely, for a set $$X$$ and a binary relation $$R$$ on $$X$$, I would call the $$R$$-induced topology as the topology generated by sets of the form:

$$l_R(x)=\{ y\in X: (y,x)\in R \}$$ and $$r_R(x)=\{ y\in X: (x,y)\in R\}$$

• GO (generalized order) spaces might interest you. Commented Jan 1, 2019 at 2:10
• This seems to be a generalization in a different direction. I'm interested in a partial order relation, since for example it induces (I think) the topology on $\mathbb{R}^d$. Commented Jan 1, 2019 at 6:31
• How is $R^d$ ordered? Commented Jan 1, 2019 at 7:28
• $\mathbb{R}^d$ is a product of linearly ordered spaces, and therefore can be given a partial order of the sort: $(a,b)<(c,d)$ if $a<c$ and $b<d$. This gives us the open rectangles in $\mathbb{R}^d$, which if I am not mistaken generate the standard topology on $\mathbb{R}^d$. Commented Jan 1, 2019 at 7:39
• If R equivalence relation, then topology is a pairwise disjoint collection of indiscret subspaces. If R is equality, space is discrete. Commented Jan 1, 2019 at 23:09