Relation induced topology

Asked Dec 31 '18 at 19:41

Active Jan 01 '19 at 07:25

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

$l(x)=\{ y\in X: y<x\} $ 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\} $

edited Jan 01 '19 at 07:25

asked Dec 31 '18 at 19:41

Keen-ameteur

7,663

GO (generalized order) spaces might interest you. – William Elliot Jan 01 '19 at 02:10
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 $\mathbb{R}^d$. – Keen-ameteur Jan 01 '19 at 06:31
How is $R^d$ ordered? – William Elliot Jan 01 '19 at 07:28
$\mathbb{R}^d$ 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 $\mathbb{R}^d$. – Keen-ameteur Jan 01 '19 at 07:39
According to Wikipedia, partially ordered spaces are a generalization of order topologies. I don't directly see how it coincides with your definition, though. – ComFreek Jan 01 '19 at 07:40
It does not seem to coincide with my definition. – Keen-ameteur Jan 01 '19 at 13:08
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If R equivalence relation, then topology is a pairwise disjoint collection of indiscret subspaces. If R is equality, space is discrete. – William Elliot Jan 01 '19 at 23:09

Relation induced topology

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