Why is a categorical product in Top uniquely equal to the product topology?
up vote
0
down vote
favorite
My question is NOT “why does the product topology satisfy the conditions of a categorical product in Top”.
Rather, my question is, why does it do so uniquely? I don’t require necessarily a fully rigorous proof, but an conceptual understanding.
My guess is: If we take the cartesian product between two topological spaces $X,Y$, but we endow this product with a trivial topology ${emptyset, Xtimes Y}$, then this also satisfies the categorical product conditions:
For any topological space $Z$ and continuous functions $f:Zto X, g:Zto Y$, there is a unique function $h=ftimes g:Zto Xtimes Y$. We don’t need to put any more conditions on the topology of $Xtimes Y$ for $h$ to be unique. (This is essentially because the cartesian product $Xtimes Y$ is already “set-theoretically restricted” enough to ensure uniqueness of $h$). Therefore, we can put any topology on $Xtimes Y$, so long as it makes $h$ unique for all such $(Z,f,g)$. One topology that does this is the product topology, but another one that does it is the topology ${emptyset,Xtimes Y }$.
What’s wrong with my argument?
EDIT: I Immediately realized that while this topology gets you a continuous $h$, it doesn’t get you continuous projections $pi_X,pi_Y$...
general-topology category-theory
asked Nov 21 at 12:21
user56834
3,14221149
|
show 1 more comment
up vote
0
down vote
favorite
My question is NOT “why does the product topology satisfy the conditions of a categorical product in Top”.
Rather, my question is, why does it do so uniquely? I don’t require necessarily a fully rigorous proof, but an conceptual understanding.
My guess is: If we take the cartesian product between two topological spaces $X,Y$, but we endow this product with a trivial topology ${emptyset, Xtimes Y}$, then this also satisfies the categorical product conditions:
For any topological space $Z$ and continuous functions $f:Zto X, g:Zto Y$, there is a unique function $h=ftimes g:Zto Xtimes Y$. We don’t need to put any more conditions on the topology of $Xtimes Y$ for $h$ to be unique. (This is essentially because the cartesian product $Xtimes Y$ is already “set-theoretically restricted” enough to ensure uniqueness of $h$). Therefore, we can put any topology on $Xtimes Y$, so long as it makes $h$ unique for all such $(Z,f,g)$. One topology that does this is the product topology, but another one that does it is the topology ${emptyset,Xtimes Y }$.
What’s wrong with my argument?
EDIT: I Immediately realized that while this topology gets you a continuous $h$, it doesn’t get you continuous projections $pi_X,pi_Y$...
general-topology category-theory
asked Nov 21 at 12:21
user56834
3,14221149
So basically you answered the question yourself. So what else do you need?
– freakish
Nov 21 at 12:25
2
@freakish, yes you’re right, lol. What should I do? Delete the question? What’s the standard practice?
– user56834
Nov 21 at 12:26
1
I guess deleting the question is fine.
– freakish
Nov 21 at 12:27
3
The general answer is that products are always determined up to isomorphism in their category.
– Kevin Carlson
Nov 21 at 15:54
1
More generally, limits are unique up to isomorphism, where they exist, and the product is an example of a limit. I second @MusaAl-hassy: you shouldn't delete the question, as it may be useful when others search similar questions. Instead, you can add an answer yourself to this!
– B. Mehta
Nov 22 at 17:12
|
show 1 more comment
up vote
0
down vote
favorite
up vote
0
down vote
favorite
My question is NOT “why does the product topology satisfy the conditions of a categorical product in Top”.
Rather, my question is, why does it do so uniquely? I don’t require necessarily a fully rigorous proof, but an conceptual understanding.
My guess is: If we take the cartesian product between two topological spaces $X,Y$, but we endow this product with a trivial topology ${emptyset, Xtimes Y}$, then this also satisfies the categorical product conditions:
For any topological space $Z$ and continuous functions $f:Zto X, g:Zto Y$, there is a unique function $h=ftimes g:Zto Xtimes Y$. We don’t need to put any more conditions on the topology of $Xtimes Y$ for $h$ to be unique. (This is essentially because the cartesian product $Xtimes Y$ is already “set-theoretically restricted” enough to ensure uniqueness of $h$). Therefore, we can put any topology on $Xtimes Y$, so long as it makes $h$ unique for all such $(Z,f,g)$. One topology that does this is the product topology, but another one that does it is the topology ${emptyset,Xtimes Y }$.
What’s wrong with my argument?
EDIT: I Immediately realized that while this topology gets you a continuous $h$, it doesn’t get you continuous projections $pi_X,pi_Y$...
general-topology category-theory
asked Nov 21 at 12:21
user56834
3,14221149
My question is NOT “why does the product topology satisfy the conditions of a categorical product in Top”.
Rather, my question is, why does it do so uniquely? I don’t require necessarily a fully rigorous proof, but an conceptual understanding.
My guess is: If we take the cartesian product between two topological spaces $X,Y$, but we endow this product with a trivial topology ${emptyset, Xtimes Y}$, then this also satisfies the categorical product conditions:
For any topological space $Z$ and continuous functions $f:Zto X, g:Zto Y$, there is a unique function $h=ftimes g:Zto Xtimes Y$. We don’t need to put any more conditions on the topology of $Xtimes Y$ for $h$ to be unique. (This is essentially because the cartesian product $Xtimes Y$ is already “set-theoretically restricted” enough to ensure uniqueness of $h$). Therefore, we can put any topology on $Xtimes Y$, so long as it makes $h$ unique for all such $(Z,f,g)$. One topology that does this is the product topology, but another one that does it is the topology ${emptyset,Xtimes Y }$.
What’s wrong with my argument?
EDIT: I Immediately realized that while this topology gets you a continuous $h$, it doesn’t get you continuous projections $pi_X,pi_Y$...
general-topology category-theory
general-topology category-theory
asked Nov 21 at 12:21
user56834
3,14221149
asked Nov 21 at 12:21
user56834
3,14221149
asked Nov 21 at 12:21
user56834
3,14221149
asked Nov 21 at 12:21
user56834
3,14221149
asked Nov 21 at 12:21
user56834
3,14221149
3,14221149
So basically you answered the question yourself. So what else do you need?
– freakish
Nov 21 at 12:25
2
@freakish, yes you’re right, lol. What should I do? Delete the question? What’s the standard practice?
– user56834
Nov 21 at 12:26
1
I guess deleting the question is fine.
– freakish
Nov 21 at 12:27
3
The general answer is that products are always determined up to isomorphism in their category.
– Kevin Carlson
Nov 21 at 15:54
1
More generally, limits are unique up to isomorphism, where they exist, and the product is an example of a limit. I second @MusaAl-hassy: you shouldn't delete the question, as it may be useful when others search similar questions. Instead, you can add an answer yourself to this!
– B. Mehta
Nov 22 at 17:12
|
show 1 more comment
So basically you answered the question yourself. So what else do you need?
– freakish
Nov 21 at 12:25
2
@freakish, yes you’re right, lol. What should I do? Delete the question? What’s the standard practice?
– user56834
Nov 21 at 12:26
1
I guess deleting the question is fine.
– freakish
Nov 21 at 12:27
3
The general answer is that products are always determined up to isomorphism in their category.
– Kevin Carlson
Nov 21 at 15:54
1
More generally, limits are unique up to isomorphism, where they exist, and the product is an example of a limit. I second @MusaAl-hassy: you shouldn't delete the question, as it may be useful when others search similar questions. Instead, you can add an answer yourself to this!
– B. Mehta
Nov 22 at 17:12
So basically you answered the question yourself. So what else do you need?
– freakish
Nov 21 at 12:25
So basically you answered the question yourself. So what else do you need?
– freakish
Nov 21 at 12:25
2
2
@freakish, yes you’re right, lol. What should I do? Delete the question? What’s the standard practice?
– user56834
Nov 21 at 12:26
@freakish, yes you’re right, lol. What should I do? Delete the question? What’s the standard practice?
– user56834
Nov 21 at 12:26
1
1
I guess deleting the question is fine.
– freakish
Nov 21 at 12:27
I guess deleting the question is fine.
– freakish
Nov 21 at 12:27
3
3
The general answer is that products are always determined up to isomorphism in their category.
– Kevin Carlson
Nov 21 at 15:54
The general answer is that products are always determined up to isomorphism in their category.
– Kevin Carlson
Nov 21 at 15:54
1
1
More generally, limits are unique up to isomorphism, where they exist, and the product is an example of a limit. I second @MusaAl-hassy: you shouldn't delete the question, as it may be useful when others search similar questions. Instead, you can add an answer yourself to this!
– B. Mehta
Nov 22 at 17:12
More generally, limits are unique up to isomorphism, where they exist, and the product is an example of a limit. I second @MusaAl-hassy: you shouldn't delete the question, as it may be useful when others search similar questions. Instead, you can add an answer yourself to this!
– B. Mehta
Nov 22 at 17:12
|
show 1 more comment
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3007653%2fwhy-is-a-categorical-product-in-top-uniquely-equal-to-the-product-topology%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
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
Required, but never shown
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
Required, but never shown
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
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
So basically you answered the question yourself. So what else do you need?
– freakish
Nov 21 at 12:25
2
@freakish, yes you’re right, lol. What should I do? Delete the question? What’s the standard practice?
– user56834
Nov 21 at 12:26
1
I guess deleting the question is fine.
– freakish
Nov 21 at 12:27
3
The general answer is that products are always determined up to isomorphism in their category.
– Kevin Carlson
Nov 21 at 15:54
1
More generally, limits are unique up to isomorphism, where they exist, and the product is an example of a limit. I second @MusaAl-hassy: you shouldn't delete the question, as it may be useful when others search similar questions. Instead, you can add an answer yourself to this!
– B. Mehta
Nov 22 at 17:12