Is there a $D$- chain between two point of a connected component in an uniform space?
$begingroup$
A uniform space $(X, U)$ is said to be uniformly connected if every uniformly continuous map
of the space into a discrete space is a constant map also a topological space is connected if and
only if the only mapping of the space into a discrete space is a constant mapping. This implies
that if uniform space $(X, mathcal{U})$ is connected, then for each pair $x,yin X$ and each $Uinmathcal{U}$, there is
an integer $n$ such that $(x, y)in U^n$.
Let $Asubseteq X$ is a connected component ($X$ is not necessary connected). Let $x,yin A$ and $Dinmathcal{U}$ be given. Is there an integer $n$ such that $(x, y)in U^n$?
Please help me to know it.
Thanks
general-topology uniform-spaces
asked Dec 8 '18 at 12:30
user479859user479859
756
$endgroup$
add a comment |
$begingroup$
A uniform space $(X, U)$ is said to be uniformly connected if every uniformly continuous map
of the space into a discrete space is a constant map also a topological space is connected if and
only if the only mapping of the space into a discrete space is a constant mapping. This implies
that if uniform space $(X, mathcal{U})$ is connected, then for each pair $x,yin X$ and each $Uinmathcal{U}$, there is
an integer $n$ such that $(x, y)in U^n$.
Let $Asubseteq X$ is a connected component ($X$ is not necessary connected). Let $x,yin A$ and $Dinmathcal{U}$ be given. Is there an integer $n$ such that $(x, y)in U^n$?
Please help me to know it.
Thanks
general-topology uniform-spaces
asked Dec 8 '18 at 12:30
user479859user479859
756
$endgroup$
1
$begingroup$
Yes, if you mean uniform-connected component.
$endgroup$
– Henno Brandsma
Dec 8 '18 at 18:47
$begingroup$
@HennoBrandsma,Thanks. In a paper, author claimed that uniformity generated by the subsets $D_n$( $ninmathbb{N}$) of $mathbb{Z}times mathbb{Z}$, where $(x,y)in D_n$ if and only if $xequiv y$ mod $2^n$, has at least one non-trivial connected component, but this is not clear for me. Please help me to know it.
$endgroup$
– user479859
Dec 8 '18 at 20:45
add a comment |
$begingroup$
A uniform space $(X, U)$ is said to be uniformly connected if every uniformly continuous map
of the space into a discrete space is a constant map also a topological space is connected if and
only if the only mapping of the space into a discrete space is a constant mapping. This implies
that if uniform space $(X, mathcal{U})$ is connected, then for each pair $x,yin X$ and each $Uinmathcal{U}$, there is
an integer $n$ such that $(x, y)in U^n$.
Let $Asubseteq X$ is a connected component ($X$ is not necessary connected). Let $x,yin A$ and $Dinmathcal{U}$ be given. Is there an integer $n$ such that $(x, y)in U^n$?
Please help me to know it.
Thanks
general-topology uniform-spaces
asked Dec 8 '18 at 12:30
user479859user479859
756
$endgroup$
A uniform space $(X, U)$ is said to be uniformly connected if every uniformly continuous map
of the space into a discrete space is a constant map also a topological space is connected if and
only if the only mapping of the space into a discrete space is a constant mapping. This implies
that if uniform space $(X, mathcal{U})$ is connected, then for each pair $x,yin X$ and each $Uinmathcal{U}$, there is
an integer $n$ such that $(x, y)in U^n$.
Let $Asubseteq X$ is a connected component ($X$ is not necessary connected). Let $x,yin A$ and $Dinmathcal{U}$ be given. Is there an integer $n$ such that $(x, y)in U^n$?
Please help me to know it.
Thanks
general-topology uniform-spaces
general-topology uniform-spaces
asked Dec 8 '18 at 12:30
user479859user479859
756
asked Dec 8 '18 at 12:30
user479859user479859
756
asked Dec 8 '18 at 12:30
user479859user479859
756
asked Dec 8 '18 at 12:30
user479859user479859
756
asked Dec 8 '18 at 12:30
user479859user479859
756
756
1
$begingroup$
Yes, if you mean uniform-connected component.
$endgroup$
– Henno Brandsma
Dec 8 '18 at 18:47
$begingroup$
@HennoBrandsma,Thanks. In a paper, author claimed that uniformity generated by the subsets $D_n$( $ninmathbb{N}$) of $mathbb{Z}times mathbb{Z}$, where $(x,y)in D_n$ if and only if $xequiv y$ mod $2^n$, has at least one non-trivial connected component, but this is not clear for me. Please help me to know it.
$endgroup$
– user479859
Dec 8 '18 at 20:45
add a comment |
1
$begingroup$
Yes, if you mean uniform-connected component.
$endgroup$
– Henno Brandsma
Dec 8 '18 at 18:47
$begingroup$
@HennoBrandsma,Thanks. In a paper, author claimed that uniformity generated by the subsets $D_n$( $ninmathbb{N}$) of $mathbb{Z}times mathbb{Z}$, where $(x,y)in D_n$ if and only if $xequiv y$ mod $2^n$, has at least one non-trivial connected component, but this is not clear for me. Please help me to know it.
$endgroup$
– user479859
Dec 8 '18 at 20:45
1
1
$begingroup$
Yes, if you mean uniform-connected component.
$endgroup$
– Henno Brandsma
Dec 8 '18 at 18:47
$begingroup$
Yes, if you mean uniform-connected component.
$endgroup$
– Henno Brandsma
Dec 8 '18 at 18:47
$begingroup$
@HennoBrandsma,Thanks. In a paper, author claimed that uniformity generated by the subsets $D_n$( $ninmathbb{N}$) of $mathbb{Z}times mathbb{Z}$, where $(x,y)in D_n$ if and only if $xequiv y$ mod $2^n$, has at least one non-trivial connected component, but this is not clear for me. Please help me to know it.
$endgroup$
– user479859
Dec 8 '18 at 20:45
$begingroup$
@HennoBrandsma,Thanks. In a paper, author claimed that uniformity generated by the subsets $D_n$( $ninmathbb{N}$) of $mathbb{Z}times mathbb{Z}$, where $(x,y)in D_n$ if and only if $xequiv y$ mod $2^n$, has at least one non-trivial connected component, but this is not clear for me. Please help me to know it.
$endgroup$
– user479859
Dec 8 '18 at 20:45
add a comment |
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$begingroup$
Yes, if you mean uniform-connected component.
$endgroup$
– Henno Brandsma
Dec 8 '18 at 18:47
$begingroup$
@HennoBrandsma,Thanks. In a paper, author claimed that uniformity generated by the subsets $D_n$( $ninmathbb{N}$) of $mathbb{Z}times mathbb{Z}$, where $(x,y)in D_n$ if and only if $xequiv y$ mod $2^n$, has at least one non-trivial connected component, but this is not clear for me. Please help me to know it.
$endgroup$
– user479859
Dec 8 '18 at 20:45