On a certain topology on the set of continuous functions between topological spaces

Clash Royale CLAN TAG#URR8PPP

up vote
2
down vote

favorite

1

Let $A,X,Y$ be topological spaces. Given a function $g: A times X to Y$, we can make a corresponding function $bar g: A to Y^X$ as $bar g(a) (x)=g(a,x), forall a in A, x in X$, and vice-versa. The "exponential topology" on $C(X,Y)$ (the set of all continuous functions from $X$ to $Y$) is a topology on it such that for every topological space $A$, $g in C(Atimes X,Y) iff bar g in C(A,C(X,Y))$ .

I can show that if an exponential topology on $C(X,Y)$ exists then it is unique.

My question is: Does there exists an exponential topology on $C(X,Y)$ when $X=Y=[0,1]$ ?

general-topology continuity

share|cite|improve this question

asked Jul 28 at 21:50

user521337

606

add a comment | 

up vote
2
down vote

favorite

1

Let $A,X,Y$ be topological spaces. Given a function $g: A times X to Y$, we can make a corresponding function $bar g: A to Y^X$ as $bar g(a) (x)=g(a,x), forall a in A, x in X$, and vice-versa. The "exponential topology" on $C(X,Y)$ (the set of all continuous functions from $X$ to $Y$) is a topology on it such that for every topological space $A$, $g in C(Atimes X,Y) iff bar g in C(A,C(X,Y))$ .

I can show that if an exponential topology on $C(X,Y)$ exists then it is unique.

My question is: Does there exists an exponential topology on $C(X,Y)$ when $X=Y=[0,1]$ ?

general-topology continuity

share|cite|improve this question

asked Jul 28 at 21:50

user521337

606

add a comment | 

up vote
2
down vote

favorite

1

up vote
2
down vote

favorite

1

1

Let $A,X,Y$ be topological spaces. Given a function $g: A times X to Y$, we can make a corresponding function $bar g: A to Y^X$ as $bar g(a) (x)=g(a,x), forall a in A, x in X$, and vice-versa. The "exponential topology" on $C(X,Y)$ (the set of all continuous functions from $X$ to $Y$) is a topology on it such that for every topological space $A$, $g in C(Atimes X,Y) iff bar g in C(A,C(X,Y))$ .

I can show that if an exponential topology on $C(X,Y)$ exists then it is unique.

My question is: Does there exists an exponential topology on $C(X,Y)$ when $X=Y=[0,1]$ ?

general-topology continuity

share|cite|improve this question

asked Jul 28 at 21:50

user521337

606

Let $A,X,Y$ be topological spaces. Given a function $g: A times X to Y$, we can make a corresponding function $bar g: A to Y^X$ as $bar g(a) (x)=g(a,x), forall a in A, x in X$, and vice-versa. The "exponential topology" on $C(X,Y)$ (the set of all continuous functions from $X$ to $Y$) is a topology on it such that for every topological space $A$, $g in C(Atimes X,Y) iff bar g in C(A,C(X,Y))$ .

I can show that if an exponential topology on $C(X,Y)$ exists then it is unique.

My question is: Does there exists an exponential topology on $C(X,Y)$ when $X=Y=[0,1]$ ?

general-topology continuity

share|cite|improve this question

asked Jul 28 at 21:50

user521337

606

share|cite|improve this question

share|cite|improve this question

asked Jul 28 at 21:50

user521337

606

asked Jul 28 at 21:50

user521337

606

asked Jul 28 at 21:50

user521337

606

606

add a comment | 

add a comment | 

1 Answer
1

active

oldest

votes

up vote
1
down vote

You should consult any book on general topology which covers function spaces, for example

Engelking, Ryszard. "General topology." (1989)

The correspondence $g mapsto overlineg$ is defined for all (not necessarily continuous) functions $g : A times X to Y$. It is called the exponential map $Lambda : mathcalF(A times X, Y) to mathcalF(A, mathcalF(X,Y))$. Here, $mathcalF(M,N)$ denotes the set of all functions $M to N$. If we restrict to continuous functions, we shall consider topologies $mathfrakT_X,Y$ on $C(X,Y)$. Let us define

(1) $mathfrakT_X,Y$ is proper if $Lambda(C(A times X,Y) subset C(A, (C(X,Y),mathfrakT_X,Y))$ for all $A$.

(2) $mathfrakT_X,Y$ is admissible if $Lambda^-1(C(A, (C(X,Y),mathfrakT_X,Y))) subset C(A times X,Y)$ for all $A$.

(3) $mathfrakT_X,Y$ is acceptable if it is both proper and admissible (this is what you call exponential topology).

It is a well-known fact that there exist at most one acceptable topology on $C(X,Y)$. Usually function spaces are endowed with the compact-open topology and it is well-known that the compact-open topology is

(a) proper for all spaces $X,Y$,

(b) admissible for all locally compact spaces $X$ and all spaces $Y$.

This answers your question in the affirmative.

share|cite|improve this answer

answered Jul 29 at 0:07

Paul Frost

3,593420

add a comment | 

Your Answer

StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: false,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);

 

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2865600%2fon-a-certain-topology-on-the-set-of-continuous-functions-between-topological-spa%23new-answer', 'question_page');

);

Post as a guest

Name Email

1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
1
down vote

You should consult any book on general topology which covers function spaces, for example

Engelking, Ryszard. "General topology." (1989)

The correspondence $g mapsto overlineg$ is defined for all (not necessarily continuous) functions $g : A times X to Y$. It is called the exponential map $Lambda : mathcalF(A times X, Y) to mathcalF(A, mathcalF(X,Y))$. Here, $mathcalF(M,N)$ denotes the set of all functions $M to N$. If we restrict to continuous functions, we shall consider topologies $mathfrakT_X,Y$ on $C(X,Y)$. Let us define

(1) $mathfrakT_X,Y$ is proper if $Lambda(C(A times X,Y) subset C(A, (C(X,Y),mathfrakT_X,Y))$ for all $A$.

(2) $mathfrakT_X,Y$ is admissible if $Lambda^-1(C(A, (C(X,Y),mathfrakT_X,Y))) subset C(A times X,Y)$ for all $A$.

(3) $mathfrakT_X,Y$ is acceptable if it is both proper and admissible (this is what you call exponential topology).

It is a well-known fact that there exist at most one acceptable topology on $C(X,Y)$. Usually function spaces are endowed with the compact-open topology and it is well-known that the compact-open topology is

(a) proper for all spaces $X,Y$,

(b) admissible for all locally compact spaces $X$ and all spaces $Y$.

This answers your question in the affirmative.

share|cite|improve this answer

answered Jul 29 at 0:07

Paul Frost

3,593420

add a comment | 

up vote
1
down vote

You should consult any book on general topology which covers function spaces, for example

Engelking, Ryszard. "General topology." (1989)

The correspondence $g mapsto overlineg$ is defined for all (not necessarily continuous) functions $g : A times X to Y$. It is called the exponential map $Lambda : mathcalF(A times X, Y) to mathcalF(A, mathcalF(X,Y))$. Here, $mathcalF(M,N)$ denotes the set of all functions $M to N$. If we restrict to continuous functions, we shall consider topologies $mathfrakT_X,Y$ on $C(X,Y)$. Let us define

(1) $mathfrakT_X,Y$ is proper if $Lambda(C(A times X,Y) subset C(A, (C(X,Y),mathfrakT_X,Y))$ for all $A$.

(2) $mathfrakT_X,Y$ is admissible if $Lambda^-1(C(A, (C(X,Y),mathfrakT_X,Y))) subset C(A times X,Y)$ for all $A$.

(3) $mathfrakT_X,Y$ is acceptable if it is both proper and admissible (this is what you call exponential topology).

It is a well-known fact that there exist at most one acceptable topology on $C(X,Y)$. Usually function spaces are endowed with the compact-open topology and it is well-known that the compact-open topology is

(a) proper for all spaces $X,Y$,

(b) admissible for all locally compact spaces $X$ and all spaces $Y$.

This answers your question in the affirmative.

share|cite|improve this answer

answered Jul 29 at 0:07

Paul Frost

3,593420

add a comment | 

up vote
1
down vote

up vote
1
down vote

You should consult any book on general topology which covers function spaces, for example

Engelking, Ryszard. "General topology." (1989)

The correspondence $g mapsto overlineg$ is defined for all (not necessarily continuous) functions $g : A times X to Y$. It is called the exponential map $Lambda : mathcalF(A times X, Y) to mathcalF(A, mathcalF(X,Y))$. Here, $mathcalF(M,N)$ denotes the set of all functions $M to N$. If we restrict to continuous functions, we shall consider topologies $mathfrakT_X,Y$ on $C(X,Y)$. Let us define

(1) $mathfrakT_X,Y$ is proper if $Lambda(C(A times X,Y) subset C(A, (C(X,Y),mathfrakT_X,Y))$ for all $A$.

(2) $mathfrakT_X,Y$ is admissible if $Lambda^-1(C(A, (C(X,Y),mathfrakT_X,Y))) subset C(A times X,Y)$ for all $A$.

(3) $mathfrakT_X,Y$ is acceptable if it is both proper and admissible (this is what you call exponential topology).

It is a well-known fact that there exist at most one acceptable topology on $C(X,Y)$. Usually function spaces are endowed with the compact-open topology and it is well-known that the compact-open topology is

(a) proper for all spaces $X,Y$,

(b) admissible for all locally compact spaces $X$ and all spaces $Y$.

This answers your question in the affirmative.

share|cite|improve this answer

answered Jul 29 at 0:07

Paul Frost

3,593420

You should consult any book on general topology which covers function spaces, for example

Engelking, Ryszard. "General topology." (1989)

The correspondence $g mapsto overlineg$ is defined for all (not necessarily continuous) functions $g : A times X to Y$. It is called the exponential map $Lambda : mathcalF(A times X, Y) to mathcalF(A, mathcalF(X,Y))$. Here, $mathcalF(M,N)$ denotes the set of all functions $M to N$. If we restrict to continuous functions, we shall consider topologies $mathfrakT_X,Y$ on $C(X,Y)$. Let us define

(1) $mathfrakT_X,Y$ is proper if $Lambda(C(A times X,Y) subset C(A, (C(X,Y),mathfrakT_X,Y))$ for all $A$.

(2) $mathfrakT_X,Y$ is admissible if $Lambda^-1(C(A, (C(X,Y),mathfrakT_X,Y))) subset C(A times X,Y)$ for all $A$.

(3) $mathfrakT_X,Y$ is acceptable if it is both proper and admissible (this is what you call exponential topology).

It is a well-known fact that there exist at most one acceptable topology on $C(X,Y)$. Usually function spaces are endowed with the compact-open topology and it is well-known that the compact-open topology is

(a) proper for all spaces $X,Y$,

(b) admissible for all locally compact spaces $X$ and all spaces $Y$.

This answers your question in the affirmative.

share|cite|improve this answer

answered Jul 29 at 0:07

Paul Frost

3,593420

share|cite|improve this answer

share|cite|improve this answer

answered Jul 29 at 0:07

Paul Frost

3,593420

answered Jul 29 at 0:07

Paul Frost

3,593420

answered Jul 29 at 0:07

Paul Frost

3,593420

3,593420

add a comment | 

add a comment | 

 

draft saved

draft discarded

 

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2865600%2fon-a-certain-topology-on-the-set-of-continuous-functions-between-topological-spa%23new-answer', 'question_page');

);

Post as a guest

Name Email

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

Name Email

Name

Email

Name Email

Name

Email