For closed set $S$ in uniform space $(X, mathcal{U})$, with $Ssubseteq W$, is there $Uin mathcal{U}$ such...
$begingroup$
Let $(X, mathcal{U})$ be a compact, Hausdorff uniform space and $Ssubseteq X$ be a closed set with $Ssubseteq W$, where $Wsubseteq X$ is an open set in $X$.
Let $U[x]={y: (x, y)in U}$ and $U[S]=cup_{xin S}U[x]$.
Is there $Uin mathcal{U}$ such that $Ssubseteq U[S]subseteq W$?
general-topology uniform-spaces
edited Dec 24 '18 at 8:35
Nosrati
26.5k62354
asked Dec 24 '18 at 8:28
user479859user479859
987
$endgroup$
add a comment |
$begingroup$
Let $(X, mathcal{U})$ be a compact, Hausdorff uniform space and $Ssubseteq X$ be a closed set with $Ssubseteq W$, where $Wsubseteq X$ is an open set in $X$.
Let $U[x]={y: (x, y)in U}$ and $U[S]=cup_{xin S}U[x]$.
Is there $Uin mathcal{U}$ such that $Ssubseteq U[S]subseteq W$?
general-topology uniform-spaces
edited Dec 24 '18 at 8:35
Nosrati
26.5k62354
asked Dec 24 '18 at 8:28
user479859user479859
987
$endgroup$
$begingroup$
Cover $S$ by finitely many $U_i[s]$, $s in S$ that all sit inside $W$.
$endgroup$
– Henno Brandsma
Dec 24 '18 at 8:44
$begingroup$
@HennoBrandsma, Yes it is true that there is finitely $U_i[s]$ that is cover for $S$ and all sit inside $W$, but I need one entorage $Uin mathcal{U}$ with $Ssubseteq U[S]subseteq W$.
$endgroup$
– user479859
Dec 24 '18 at 9:52
add a comment |
$begingroup$
Let $(X, mathcal{U})$ be a compact, Hausdorff uniform space and $Ssubseteq X$ be a closed set with $Ssubseteq W$, where $Wsubseteq X$ is an open set in $X$.
Let $U[x]={y: (x, y)in U}$ and $U[S]=cup_{xin S}U[x]$.
Is there $Uin mathcal{U}$ such that $Ssubseteq U[S]subseteq W$?
general-topology uniform-spaces
edited Dec 24 '18 at 8:35
Nosrati
26.5k62354
asked Dec 24 '18 at 8:28
user479859user479859
987
$endgroup$
Let $(X, mathcal{U})$ be a compact, Hausdorff uniform space and $Ssubseteq X$ be a closed set with $Ssubseteq W$, where $Wsubseteq X$ is an open set in $X$.
Let $U[x]={y: (x, y)in U}$ and $U[S]=cup_{xin S}U[x]$.
Is there $Uin mathcal{U}$ such that $Ssubseteq U[S]subseteq W$?
general-topology uniform-spaces
general-topology uniform-spaces
edited Dec 24 '18 at 8:35
Nosrati
26.5k62354
asked Dec 24 '18 at 8:28
user479859user479859
987
edited Dec 24 '18 at 8:35
Nosrati
26.5k62354
asked Dec 24 '18 at 8:28
user479859user479859
987
edited Dec 24 '18 at 8:35
Nosrati
26.5k62354
edited Dec 24 '18 at 8:35
Nosrati
26.5k62354
edited Dec 24 '18 at 8:35
Nosrati
26.5k62354
26.5k62354
asked Dec 24 '18 at 8:28
user479859user479859
987
asked Dec 24 '18 at 8:28
user479859user479859
987
asked Dec 24 '18 at 8:28
user479859user479859
987
987
$begingroup$
Cover $S$ by finitely many $U_i[s]$, $s in S$ that all sit inside $W$.
$endgroup$
– Henno Brandsma
Dec 24 '18 at 8:44
$begingroup$
@HennoBrandsma, Yes it is true that there is finitely $U_i[s]$ that is cover for $S$ and all sit inside $W$, but I need one entorage $Uin mathcal{U}$ with $Ssubseteq U[S]subseteq W$.
$endgroup$
– user479859
Dec 24 '18 at 9:52
add a comment |
$begingroup$
Cover $S$ by finitely many $U_i[s]$, $s in S$ that all sit inside $W$.
$endgroup$
– Henno Brandsma
Dec 24 '18 at 8:44
$begingroup$
@HennoBrandsma, Yes it is true that there is finitely $U_i[s]$ that is cover for $S$ and all sit inside $W$, but I need one entorage $Uin mathcal{U}$ with $Ssubseteq U[S]subseteq W$.
$endgroup$
– user479859
Dec 24 '18 at 9:52
$begingroup$
Cover $S$ by finitely many $U_i[s]$, $s in S$ that all sit inside $W$.
$endgroup$
– Henno Brandsma
Dec 24 '18 at 8:44
$begingroup$
Cover $S$ by finitely many $U_i[s]$, $s in S$ that all sit inside $W$.
$endgroup$
– Henno Brandsma
Dec 24 '18 at 8:44
$begingroup$
@HennoBrandsma, Yes it is true that there is finitely $U_i[s]$ that is cover for $S$ and all sit inside $W$, but I need one entorage $Uin mathcal{U}$ with $Ssubseteq U[S]subseteq W$.
$endgroup$
– user479859
Dec 24 '18 at 9:52
$begingroup$
@HennoBrandsma, Yes it is true that there is finitely $U_i[s]$ that is cover for $S$ and all sit inside $W$, but I need one entorage $Uin mathcal{U}$ with $Ssubseteq U[S]subseteq W$.
$endgroup$
– user479859
Dec 24 '18 at 9:52
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Is there $Uin mathcal{U}$ such that $Ssubseteq U[S]subseteq W$?
Yes. For each $xin S$ pick a symmetric entourage $U_xinmathcal U$ such that $U_x^2[x]subseteq W$. Since the set $S$ is compact, there exists a finite subset $F$ of $S$ such that $Ssubseteqbigcup{U_x[x]: xin F} $. Put $U=bigcap {U_x:xin F}$. Then
$$U[S]subseteq Uleft[bigcup{U_x[x]: xin F}right] subseteq bigcup{ U[U_x[x]]: xin F}
subseteq bigcup{ U^2_x[x]: xin F}]subseteq W.$$
answered Dec 24 '18 at 21:45
Alex RavskyAlex Ravsky
42.8k32483
$endgroup$
add a comment |
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1 Answer
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1 Answer
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oldest
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$begingroup$
Is there $Uin mathcal{U}$ such that $Ssubseteq U[S]subseteq W$?
Yes. For each $xin S$ pick a symmetric entourage $U_xinmathcal U$ such that $U_x^2[x]subseteq W$. Since the set $S$ is compact, there exists a finite subset $F$ of $S$ such that $Ssubseteqbigcup{U_x[x]: xin F} $. Put $U=bigcap {U_x:xin F}$. Then
$$U[S]subseteq Uleft[bigcup{U_x[x]: xin F}right] subseteq bigcup{ U[U_x[x]]: xin F}
subseteq bigcup{ U^2_x[x]: xin F}]subseteq W.$$
answered Dec 24 '18 at 21:45
Alex RavskyAlex Ravsky
42.8k32483
$endgroup$
add a comment |
$begingroup$
Is there $Uin mathcal{U}$ such that $Ssubseteq U[S]subseteq W$?
Yes. For each $xin S$ pick a symmetric entourage $U_xinmathcal U$ such that $U_x^2[x]subseteq W$. Since the set $S$ is compact, there exists a finite subset $F$ of $S$ such that $Ssubseteqbigcup{U_x[x]: xin F} $. Put $U=bigcap {U_x:xin F}$. Then
$$U[S]subseteq Uleft[bigcup{U_x[x]: xin F}right] subseteq bigcup{ U[U_x[x]]: xin F}
subseteq bigcup{ U^2_x[x]: xin F}]subseteq W.$$
answered Dec 24 '18 at 21:45
Alex RavskyAlex Ravsky
42.8k32483
$endgroup$
add a comment |
$begingroup$
Is there $Uin mathcal{U}$ such that $Ssubseteq U[S]subseteq W$?
Yes. For each $xin S$ pick a symmetric entourage $U_xinmathcal U$ such that $U_x^2[x]subseteq W$. Since the set $S$ is compact, there exists a finite subset $F$ of $S$ such that $Ssubseteqbigcup{U_x[x]: xin F} $. Put $U=bigcap {U_x:xin F}$. Then
$$U[S]subseteq Uleft[bigcup{U_x[x]: xin F}right] subseteq bigcup{ U[U_x[x]]: xin F}
subseteq bigcup{ U^2_x[x]: xin F}]subseteq W.$$
answered Dec 24 '18 at 21:45
Alex RavskyAlex Ravsky
42.8k32483
$endgroup$
Is there $Uin mathcal{U}$ such that $Ssubseteq U[S]subseteq W$?
Yes. For each $xin S$ pick a symmetric entourage $U_xinmathcal U$ such that $U_x^2[x]subseteq W$. Since the set $S$ is compact, there exists a finite subset $F$ of $S$ such that $Ssubseteqbigcup{U_x[x]: xin F} $. Put $U=bigcap {U_x:xin F}$. Then
$$U[S]subseteq Uleft[bigcup{U_x[x]: xin F}right] subseteq bigcup{ U[U_x[x]]: xin F}
subseteq bigcup{ U^2_x[x]: xin F}]subseteq W.$$
answered Dec 24 '18 at 21:45
Alex RavskyAlex Ravsky
42.8k32483
answered Dec 24 '18 at 21:45
Alex RavskyAlex Ravsky
42.8k32483
answered Dec 24 '18 at 21:45
Alex RavskyAlex Ravsky
42.8k32483
answered Dec 24 '18 at 21:45
Alex RavskyAlex Ravsky
42.8k32483
42.8k32483
add a comment |
add a comment |
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$begingroup$
Cover $S$ by finitely many $U_i[s]$, $s in S$ that all sit inside $W$.
$endgroup$
– Henno Brandsma
Dec 24 '18 at 8:44
$begingroup$
@HennoBrandsma, Yes it is true that there is finitely $U_i[s]$ that is cover for $S$ and all sit inside $W$, but I need one entorage $Uin mathcal{U}$ with $Ssubseteq U[S]subseteq W$.
$endgroup$
– user479859
Dec 24 '18 at 9:52