Uniform continuity and compactness
$begingroup$
We know that, if a function $f$ is continuous mapping from a compact metric space to another metric space say $Y$ then $f$ is uniformly continuous.
Do we have a generalization of this theorem for general topological space.
general-topology continuity compactness uniform-continuity
edited Jan 3 at 16:15
Scientifica
6,79641335
asked Jan 3 at 16:03
StammeringMathematicianStammeringMathematician
2,5911324
$endgroup$
|
show 1 more comment
$begingroup$
We know that, if a function $f$ is continuous mapping from a compact metric space to another metric space say $Y$ then $f$ is uniformly continuous.
Do we have a generalization of this theorem for general topological space.
general-topology continuity compactness uniform-continuity
edited Jan 3 at 16:15
Scientifica
6,79641335
asked Jan 3 at 16:03
StammeringMathematicianStammeringMathematician
2,5911324
$endgroup$
1
$begingroup$
I think there is none because we cannot define uniform continuity properly without the metric.
$endgroup$
– Song
Jan 3 at 16:18
$begingroup$
It's not true for non-compact spaces + see @Song 's comment
$endgroup$
– Yanko
Jan 3 at 16:18
6
$begingroup$
To talk of uniform continuity in a topological space the space need to have an additional structure, the so-called uniform structure. The spaces with this structure are called uniform spaces and yes the uniform continuity do generalize to them. For example, topological groups and compact Hausdorff are uniform spaces.
$endgroup$
– mouthetics
Jan 3 at 16:21
1
$begingroup$
@StammeringMathematician You're welcome. Happy new year.
$endgroup$
– mouthetics
Jan 3 at 16:29
1
$begingroup$
@mouthetics I also learned there is uniform structure from you comment. Thank you!
$endgroup$
– Song
Jan 3 at 16:36
|
show 1 more comment
$begingroup$
We know that, if a function $f$ is continuous mapping from a compact metric space to another metric space say $Y$ then $f$ is uniformly continuous.
Do we have a generalization of this theorem for general topological space.
general-topology continuity compactness uniform-continuity
edited Jan 3 at 16:15
Scientifica
6,79641335
asked Jan 3 at 16:03
StammeringMathematicianStammeringMathematician
2,5911324
$endgroup$
We know that, if a function $f$ is continuous mapping from a compact metric space to another metric space say $Y$ then $f$ is uniformly continuous.
Do we have a generalization of this theorem for general topological space.
general-topology continuity compactness uniform-continuity
general-topology continuity compactness uniform-continuity
edited Jan 3 at 16:15
Scientifica
6,79641335
asked Jan 3 at 16:03
StammeringMathematicianStammeringMathematician
2,5911324
edited Jan 3 at 16:15
Scientifica
6,79641335
asked Jan 3 at 16:03
StammeringMathematicianStammeringMathematician
2,5911324
edited Jan 3 at 16:15
Scientifica
6,79641335
edited Jan 3 at 16:15
Scientifica
6,79641335
edited Jan 3 at 16:15
Scientifica
6,79641335
6,79641335
asked Jan 3 at 16:03
StammeringMathematicianStammeringMathematician
2,5911324
asked Jan 3 at 16:03
StammeringMathematicianStammeringMathematician
2,5911324
asked Jan 3 at 16:03
StammeringMathematicianStammeringMathematician
2,5911324
2,5911324
1
$begingroup$
I think there is none because we cannot define uniform continuity properly without the metric.
$endgroup$
– Song
Jan 3 at 16:18
$begingroup$
It's not true for non-compact spaces + see @Song 's comment
$endgroup$
– Yanko
Jan 3 at 16:18
6
$begingroup$
To talk of uniform continuity in a topological space the space need to have an additional structure, the so-called uniform structure. The spaces with this structure are called uniform spaces and yes the uniform continuity do generalize to them. For example, topological groups and compact Hausdorff are uniform spaces.
$endgroup$
– mouthetics
Jan 3 at 16:21
1
$begingroup$
@StammeringMathematician You're welcome. Happy new year.
$endgroup$
– mouthetics
Jan 3 at 16:29
1
$begingroup$
@mouthetics I also learned there is uniform structure from you comment. Thank you!
$endgroup$
– Song
Jan 3 at 16:36
|
show 1 more comment
1
$begingroup$
I think there is none because we cannot define uniform continuity properly without the metric.
$endgroup$
– Song
Jan 3 at 16:18
$begingroup$
It's not true for non-compact spaces + see @Song 's comment
$endgroup$
– Yanko
Jan 3 at 16:18
6
$begingroup$
To talk of uniform continuity in a topological space the space need to have an additional structure, the so-called uniform structure. The spaces with this structure are called uniform spaces and yes the uniform continuity do generalize to them. For example, topological groups and compact Hausdorff are uniform spaces.
$endgroup$
– mouthetics
Jan 3 at 16:21
1
$begingroup$
@StammeringMathematician You're welcome. Happy new year.
$endgroup$
– mouthetics
Jan 3 at 16:29
1
$begingroup$
@mouthetics I also learned there is uniform structure from you comment. Thank you!
$endgroup$
– Song
Jan 3 at 16:36
1
1
$begingroup$
I think there is none because we cannot define uniform continuity properly without the metric.
$endgroup$
– Song
Jan 3 at 16:18
$begingroup$
I think there is none because we cannot define uniform continuity properly without the metric.
$endgroup$
– Song
Jan 3 at 16:18
$begingroup$
It's not true for non-compact spaces + see @Song 's comment
$endgroup$
– Yanko
Jan 3 at 16:18
$begingroup$
It's not true for non-compact spaces + see @Song 's comment
$endgroup$
– Yanko
Jan 3 at 16:18
6
6
$begingroup$
To talk of uniform continuity in a topological space the space need to have an additional structure, the so-called uniform structure. The spaces with this structure are called uniform spaces and yes the uniform continuity do generalize to them. For example, topological groups and compact Hausdorff are uniform spaces.
$endgroup$
– mouthetics
Jan 3 at 16:21
$begingroup$
To talk of uniform continuity in a topological space the space need to have an additional structure, the so-called uniform structure. The spaces with this structure are called uniform spaces and yes the uniform continuity do generalize to them. For example, topological groups and compact Hausdorff are uniform spaces.
$endgroup$
– mouthetics
Jan 3 at 16:21
1
1
$begingroup$
@StammeringMathematician You're welcome. Happy new year.
$endgroup$
– mouthetics
Jan 3 at 16:29
$begingroup$
@StammeringMathematician You're welcome. Happy new year.
$endgroup$
– mouthetics
Jan 3 at 16:29
1
1
$begingroup$
@mouthetics I also learned there is uniform structure from you comment. Thank you!
$endgroup$
– Song
Jan 3 at 16:36
$begingroup$
@mouthetics I also learned there is uniform structure from you comment. Thank you!
$endgroup$
– Song
Jan 3 at 16:36
|
show 1 more comment
1 Answer
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oldest
votes
$begingroup$
To generalize the result, you need to generalize the notions. Continuity is defined in the context of topological spaces. However, uniform continuity is not.
Here's the definition of uniform continuity for metric spaces:
Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces.
A function $f:Xto Y$ is uniformly continuous if
$$forallvarepsilon>0,existsdelta>0,forall x,yin X, d_X(x,y)<deltaimplies d_Y(f(x),f(y))<varepsilon.$$
Unlike the definition of continuity, you see that we're dealing with two variables, and you need to express the idea that "wherever $x$ and $y$ are, if they are close enough". Uniform spaces generalize this notion.
Every metric space can be given a uniform structure compatible with the distance, just like every metric space can be given a topological structure compatible with the distance. Also, every uniform space can be given a topological structure compatible with the uniform structure. So one can speak about compactness or continuity when it comes to uniform spaces. Uniform continuity of a function between two uniform spaces is defined in a similar way to continuity for topological spaces.
Your result generalizes to this:
Let $X$, $Y$ be uniform spaces and $f:Xto Y$ be a continuous function.
Suppose that $X$ is compact. Then $f$ is uniformly continuous.
I guess you'll find this result in any book about uniform spaces. For example, see Proposition 8.17 page 133 in Topologies and Uniformities.
edited Jan 3 at 19:00
Henno Brandsma
111k348119
answered Jan 3 at 16:47
ScientificaScientifica
6,79641335
$endgroup$
add a comment |
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1 Answer
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$begingroup$
To generalize the result, you need to generalize the notions. Continuity is defined in the context of topological spaces. However, uniform continuity is not.
Here's the definition of uniform continuity for metric spaces:
Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces.
A function $f:Xto Y$ is uniformly continuous if
$$forallvarepsilon>0,existsdelta>0,forall x,yin X, d_X(x,y)<deltaimplies d_Y(f(x),f(y))<varepsilon.$$
Unlike the definition of continuity, you see that we're dealing with two variables, and you need to express the idea that "wherever $x$ and $y$ are, if they are close enough". Uniform spaces generalize this notion.
Every metric space can be given a uniform structure compatible with the distance, just like every metric space can be given a topological structure compatible with the distance. Also, every uniform space can be given a topological structure compatible with the uniform structure. So one can speak about compactness or continuity when it comes to uniform spaces. Uniform continuity of a function between two uniform spaces is defined in a similar way to continuity for topological spaces.
Your result generalizes to this:
Let $X$, $Y$ be uniform spaces and $f:Xto Y$ be a continuous function.
Suppose that $X$ is compact. Then $f$ is uniformly continuous.
I guess you'll find this result in any book about uniform spaces. For example, see Proposition 8.17 page 133 in Topologies and Uniformities.
edited Jan 3 at 19:00
Henno Brandsma
111k348119
answered Jan 3 at 16:47
ScientificaScientifica
6,79641335
$endgroup$
add a comment |
$begingroup$
To generalize the result, you need to generalize the notions. Continuity is defined in the context of topological spaces. However, uniform continuity is not.
Here's the definition of uniform continuity for metric spaces:
Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces.
A function $f:Xto Y$ is uniformly continuous if
$$forallvarepsilon>0,existsdelta>0,forall x,yin X, d_X(x,y)<deltaimplies d_Y(f(x),f(y))<varepsilon.$$
Unlike the definition of continuity, you see that we're dealing with two variables, and you need to express the idea that "wherever $x$ and $y$ are, if they are close enough". Uniform spaces generalize this notion.
Every metric space can be given a uniform structure compatible with the distance, just like every metric space can be given a topological structure compatible with the distance. Also, every uniform space can be given a topological structure compatible with the uniform structure. So one can speak about compactness or continuity when it comes to uniform spaces. Uniform continuity of a function between two uniform spaces is defined in a similar way to continuity for topological spaces.
Your result generalizes to this:
Let $X$, $Y$ be uniform spaces and $f:Xto Y$ be a continuous function.
Suppose that $X$ is compact. Then $f$ is uniformly continuous.
I guess you'll find this result in any book about uniform spaces. For example, see Proposition 8.17 page 133 in Topologies and Uniformities.
edited Jan 3 at 19:00
Henno Brandsma
111k348119
answered Jan 3 at 16:47
ScientificaScientifica
6,79641335
$endgroup$
add a comment |
$begingroup$
To generalize the result, you need to generalize the notions. Continuity is defined in the context of topological spaces. However, uniform continuity is not.
Here's the definition of uniform continuity for metric spaces:
Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces.
A function $f:Xto Y$ is uniformly continuous if
$$forallvarepsilon>0,existsdelta>0,forall x,yin X, d_X(x,y)<deltaimplies d_Y(f(x),f(y))<varepsilon.$$
Unlike the definition of continuity, you see that we're dealing with two variables, and you need to express the idea that "wherever $x$ and $y$ are, if they are close enough". Uniform spaces generalize this notion.
Every metric space can be given a uniform structure compatible with the distance, just like every metric space can be given a topological structure compatible with the distance. Also, every uniform space can be given a topological structure compatible with the uniform structure. So one can speak about compactness or continuity when it comes to uniform spaces. Uniform continuity of a function between two uniform spaces is defined in a similar way to continuity for topological spaces.
Your result generalizes to this:
Let $X$, $Y$ be uniform spaces and $f:Xto Y$ be a continuous function.
Suppose that $X$ is compact. Then $f$ is uniformly continuous.
I guess you'll find this result in any book about uniform spaces. For example, see Proposition 8.17 page 133 in Topologies and Uniformities.
edited Jan 3 at 19:00
Henno Brandsma
111k348119
answered Jan 3 at 16:47
ScientificaScientifica
6,79641335
$endgroup$
To generalize the result, you need to generalize the notions. Continuity is defined in the context of topological spaces. However, uniform continuity is not.
Here's the definition of uniform continuity for metric spaces:
Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces.
A function $f:Xto Y$ is uniformly continuous if
$$forallvarepsilon>0,existsdelta>0,forall x,yin X, d_X(x,y)<deltaimplies d_Y(f(x),f(y))<varepsilon.$$
Unlike the definition of continuity, you see that we're dealing with two variables, and you need to express the idea that "wherever $x$ and $y$ are, if they are close enough". Uniform spaces generalize this notion.
Every metric space can be given a uniform structure compatible with the distance, just like every metric space can be given a topological structure compatible with the distance. Also, every uniform space can be given a topological structure compatible with the uniform structure. So one can speak about compactness or continuity when it comes to uniform spaces. Uniform continuity of a function between two uniform spaces is defined in a similar way to continuity for topological spaces.
Your result generalizes to this:
Let $X$, $Y$ be uniform spaces and $f:Xto Y$ be a continuous function.
Suppose that $X$ is compact. Then $f$ is uniformly continuous.
I guess you'll find this result in any book about uniform spaces. For example, see Proposition 8.17 page 133 in Topologies and Uniformities.
edited Jan 3 at 19:00
Henno Brandsma
111k348119
answered Jan 3 at 16:47
ScientificaScientifica
6,79641335
edited Jan 3 at 19:00
Henno Brandsma
111k348119
edited Jan 3 at 19:00
Henno Brandsma
111k348119
edited Jan 3 at 19:00
Henno Brandsma
111k348119
111k348119
answered Jan 3 at 16:47
ScientificaScientifica
6,79641335
answered Jan 3 at 16:47
ScientificaScientifica
6,79641335
answered Jan 3 at 16:47
ScientificaScientifica
6,79641335
6,79641335
add a comment |
add a comment |
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1
$begingroup$
I think there is none because we cannot define uniform continuity properly without the metric.
$endgroup$
– Song
Jan 3 at 16:18
$begingroup$
It's not true for non-compact spaces + see @Song 's comment
$endgroup$
– Yanko
Jan 3 at 16:18
6
$begingroup$
To talk of uniform continuity in a topological space the space need to have an additional structure, the so-called uniform structure. The spaces with this structure are called uniform spaces and yes the uniform continuity do generalize to them. For example, topological groups and compact Hausdorff are uniform spaces.
$endgroup$
– mouthetics
Jan 3 at 16:21
1
$begingroup$
@StammeringMathematician You're welcome. Happy new year.
$endgroup$
– mouthetics
Jan 3 at 16:29
1
$begingroup$
@mouthetics I also learned there is uniform structure from you comment. Thank you!
$endgroup$
– Song
Jan 3 at 16:36