Relation induced topology
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For an ordered space $X$ there is the term of ordered topology generated by sets of the form:
$l(x)={ yin X: y<x} $ and $r(x)={ yin 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)={ yin X: (y,x)in R } $ and $r_R(x)={ yin X: (x,y)in R} $
general-topology reference-request relations
asked Dec 31 '18 at 19:41
Keen-ameteurKeen-ameteur
1,550516
$endgroup$
|
show 2 more comments
$begingroup$
For an ordered space $X$ there is the term of ordered topology generated by sets of the form:
$l(x)={ yin X: y<x} $ and $r(x)={ yin 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)={ yin X: (y,x)in R } $ and $r_R(x)={ yin X: (x,y)in R} $
general-topology reference-request relations
asked Dec 31 '18 at 19:41
Keen-ameteurKeen-ameteur
1,550516
$endgroup$
$begingroup$
GO (generalized order) spaces might interest you.
$endgroup$
– William Elliot
Jan 1 at 2:10
$begingroup$
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 $mathbb{R}^d$.
$endgroup$
– Keen-ameteur
Jan 1 at 6:31
$begingroup$
How is $R^d$ ordered?
$endgroup$
– William Elliot
Jan 1 at 7:28
$begingroup$
$mathbb{R}^d$ 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 $mathbb{R}^d$.
$endgroup$
– Keen-ameteur
Jan 1 at 7:39
$begingroup$
According to Wikipedia, partially ordered spaces are a generalization of order topologies. I don't directly see how it coincides with your definition, though.
$endgroup$
– ComFreek
Jan 1 at 7:40
|
show 2 more comments
$begingroup$
For an ordered space $X$ there is the term of ordered topology generated by sets of the form:
$l(x)={ yin X: y<x} $ and $r(x)={ yin 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)={ yin X: (y,x)in R } $ and $r_R(x)={ yin X: (x,y)in R} $
general-topology reference-request relations
asked Dec 31 '18 at 19:41
Keen-ameteurKeen-ameteur
1,550516
$endgroup$
For an ordered space $X$ there is the term of ordered topology generated by sets of the form:
$l(x)={ yin X: y<x} $ and $r(x)={ yin 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)={ yin X: (y,x)in R } $ and $r_R(x)={ yin X: (x,y)in R} $
general-topology reference-request relations
general-topology reference-request relations
asked Dec 31 '18 at 19:41
Keen-ameteurKeen-ameteur
1,550516
asked Dec 31 '18 at 19:41
Keen-ameteurKeen-ameteur
1,550516
edited Jan 1 at 7:25
Keen-ameteur
asked Dec 31 '18 at 19:41
Keen-ameteurKeen-ameteur
1,550516
asked Dec 31 '18 at 19:41
Keen-ameteurKeen-ameteur
1,550516
asked Dec 31 '18 at 19:41
Keen-ameteurKeen-ameteur
1,550516
1,550516
$begingroup$
GO (generalized order) spaces might interest you.
$endgroup$
– William Elliot
Jan 1 at 2:10
$begingroup$
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 $mathbb{R}^d$.
$endgroup$
– Keen-ameteur
Jan 1 at 6:31
$begingroup$
How is $R^d$ ordered?
$endgroup$
– William Elliot
Jan 1 at 7:28
$begingroup$
$mathbb{R}^d$ 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 $mathbb{R}^d$.
$endgroup$
– Keen-ameteur
Jan 1 at 7:39
$begingroup$
According to Wikipedia, partially ordered spaces are a generalization of order topologies. I don't directly see how it coincides with your definition, though.
$endgroup$
– ComFreek
Jan 1 at 7:40
|
show 2 more comments
$begingroup$
GO (generalized order) spaces might interest you.
$endgroup$
– William Elliot
Jan 1 at 2:10
$begingroup$
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 $mathbb{R}^d$.
$endgroup$
– Keen-ameteur
Jan 1 at 6:31
$begingroup$
How is $R^d$ ordered?
$endgroup$
– William Elliot
Jan 1 at 7:28
$begingroup$
$mathbb{R}^d$ 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 $mathbb{R}^d$.
$endgroup$
– Keen-ameteur
Jan 1 at 7:39
$begingroup$
According to Wikipedia, partially ordered spaces are a generalization of order topologies. I don't directly see how it coincides with your definition, though.
$endgroup$
– ComFreek
Jan 1 at 7:40
$begingroup$
GO (generalized order) spaces might interest you.
$endgroup$
– William Elliot
Jan 1 at 2:10
$begingroup$
GO (generalized order) spaces might interest you.
$endgroup$
– William Elliot
Jan 1 at 2:10
$begingroup$
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 $mathbb{R}^d$.
$endgroup$
– Keen-ameteur
Jan 1 at 6:31
$begingroup$
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 $mathbb{R}^d$.
$endgroup$
– Keen-ameteur
Jan 1 at 6:31
$begingroup$
How is $R^d$ ordered?
$endgroup$
– William Elliot
Jan 1 at 7:28
$begingroup$
How is $R^d$ ordered?
$endgroup$
– William Elliot
Jan 1 at 7:28
$begingroup$
$mathbb{R}^d$ 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 $mathbb{R}^d$.
$endgroup$
– Keen-ameteur
Jan 1 at 7:39
$begingroup$
$mathbb{R}^d$ 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 $mathbb{R}^d$.
$endgroup$
– Keen-ameteur
Jan 1 at 7:39
$begingroup$
According to Wikipedia, partially ordered spaces are a generalization of order topologies. I don't directly see how it coincides with your definition, though.
$endgroup$
– ComFreek
Jan 1 at 7:40
$begingroup$
According to Wikipedia, partially ordered spaces are a generalization of order topologies. I don't directly see how it coincides with your definition, though.
$endgroup$
– ComFreek
Jan 1 at 7:40
|
show 2 more comments
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$begingroup$
GO (generalized order) spaces might interest you.
$endgroup$
– William Elliot
Jan 1 at 2:10
$begingroup$
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 $mathbb{R}^d$.
$endgroup$
– Keen-ameteur
Jan 1 at 6:31
$begingroup$
How is $R^d$ ordered?
$endgroup$
– William Elliot
Jan 1 at 7:28
$begingroup$
$mathbb{R}^d$ 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 $mathbb{R}^d$.
$endgroup$
– Keen-ameteur
Jan 1 at 7:39
$begingroup$
According to Wikipedia, partially ordered spaces are a generalization of order topologies. I don't directly see how it coincides with your definition, though.
$endgroup$
– ComFreek
Jan 1 at 7:40