Mixing
A function that is continous but non constant between two particular topological spaces [closed]
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Find a non-constant function between $X,tau_1$ and $(X,tau)$ and $(X,tau')$ where $tau={X,12,34,emptyset}$ and $tau'={X,123,12,1,emptyset}$.
$f:(X,tau)to (X,tau')$
I know that I need to find a function $f$ such that $f^{-1}(U)intau$ for all $Uin tau'$. However I am not seeing the non-constant function form.
Question:
What would you suggest as a function fulfilling the aforementioned conditions?
Thanks in advance!
general-topology examples-counterexamples
asked Feb 1 at 22:27
Pedro GomesPedro Gomes
2,0062821
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closed as unclear what you're asking by Xander Henderson, José Carlos Santos, Henno Brandsma, Shailesh, Lord Shark the Unknown Feb 2 at 7:51
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
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$begingroup$
Find a non-constant function between $X,tau_1$ and $(X,tau)$ and $(X,tau')$ where $tau={X,12,34,emptyset}$ and $tau'={X,123,12,1,emptyset}$.
$f:(X,tau)to (X,tau')$
I know that I need to find a function $f$ such that $f^{-1}(U)intau$ for all $Uin tau'$. However I am not seeing the non-constant function form.
Question:
What would you suggest as a function fulfilling the aforementioned conditions?
Thanks in advance!
general-topology examples-counterexamples
asked Feb 1 at 22:27
Pedro GomesPedro Gomes
2,0062821
$endgroup$
closed as unclear what you're asking by Xander Henderson, José Carlos Santos, Henno Brandsma, Shailesh, Lord Shark the Unknown Feb 2 at 7:51
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
1
$begingroup$
What is $tau_1$? From what space to what space does $f$ have to go? The question is unclear. $X={1,2,3,4}$?
$endgroup$
– Henno Brandsma
Feb 1 at 22:56
$begingroup$
@HennoBrandsma Please check my edit.
$endgroup$
– Pedro Gomes
Feb 1 at 22:59
1
$begingroup$
$tau_1$ should go? You still haven't defined $X$. Maths is precision! Also, don't write $12$ for ${1,2}$ etc.
$endgroup$
– Henno Brandsma
Feb 1 at 23:00
add a comment |
$begingroup$
Find a non-constant function between $X,tau_1$ and $(X,tau)$ and $(X,tau')$ where $tau={X,12,34,emptyset}$ and $tau'={X,123,12,1,emptyset}$.
$f:(X,tau)to (X,tau')$
I know that I need to find a function $f$ such that $f^{-1}(U)intau$ for all $Uin tau'$. However I am not seeing the non-constant function form.
Question:
What would you suggest as a function fulfilling the aforementioned conditions?
Thanks in advance!
general-topology examples-counterexamples
asked Feb 1 at 22:27
Pedro GomesPedro Gomes
2,0062821
$endgroup$
Find a non-constant function between $X,tau_1$ and $(X,tau)$ and $(X,tau')$ where $tau={X,12,34,emptyset}$ and $tau'={X,123,12,1,emptyset}$.
$f:(X,tau)to (X,tau')$
I know that I need to find a function $f$ such that $f^{-1}(U)intau$ for all $Uin tau'$. However I am not seeing the non-constant function form.
Question:
What would you suggest as a function fulfilling the aforementioned conditions?
Thanks in advance!
general-topology examples-counterexamples
general-topology examples-counterexamples
asked Feb 1 at 22:27
Pedro GomesPedro Gomes
2,0062821
asked Feb 1 at 22:27
Pedro GomesPedro Gomes
2,0062821
edited Feb 1 at 22:59
Pedro Gomes
asked Feb 1 at 22:27
Pedro GomesPedro Gomes
2,0062821
asked Feb 1 at 22:27
Pedro GomesPedro Gomes
2,0062821
asked Feb 1 at 22:27
Pedro GomesPedro Gomes
2,0062821
2,0062821
closed as unclear what you're asking by Xander Henderson, José Carlos Santos, Henno Brandsma, Shailesh, Lord Shark the Unknown Feb 2 at 7:51
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as unclear what you're asking by Xander Henderson, José Carlos Santos, Henno Brandsma, Shailesh, Lord Shark the Unknown Feb 2 at 7:51
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
1
$begingroup$
What is $tau_1$? From what space to what space does $f$ have to go? The question is unclear. $X={1,2,3,4}$?
$endgroup$
– Henno Brandsma
Feb 1 at 22:56
$begingroup$
@HennoBrandsma Please check my edit.
$endgroup$
– Pedro Gomes
Feb 1 at 22:59
1
$begingroup$
$tau_1$ should go? You still haven't defined $X$. Maths is precision! Also, don't write $12$ for ${1,2}$ etc.
$endgroup$
– Henno Brandsma
Feb 1 at 23:00
add a comment |
1
$begingroup$
What is $tau_1$? From what space to what space does $f$ have to go? The question is unclear. $X={1,2,3,4}$?
$endgroup$
– Henno Brandsma
Feb 1 at 22:56
$begingroup$
@HennoBrandsma Please check my edit.
$endgroup$
– Pedro Gomes
Feb 1 at 22:59
1
$begingroup$
$tau_1$ should go? You still haven't defined $X$. Maths is precision! Also, don't write $12$ for ${1,2}$ etc.
$endgroup$
– Henno Brandsma
Feb 1 at 23:00
1
1
$begingroup$
What is $tau_1$? From what space to what space does $f$ have to go? The question is unclear. $X={1,2,3,4}$?
$endgroup$
– Henno Brandsma
Feb 1 at 22:56
$begingroup$
What is $tau_1$? From what space to what space does $f$ have to go? The question is unclear. $X={1,2,3,4}$?
$endgroup$
– Henno Brandsma
Feb 1 at 22:56
$begingroup$
@HennoBrandsma Please check my edit.
$endgroup$
– Pedro Gomes
Feb 1 at 22:59
$begingroup$
@HennoBrandsma Please check my edit.
$endgroup$
– Pedro Gomes
Feb 1 at 22:59
1
1
$begingroup$
$tau_1$ should go? You still haven't defined $X$. Maths is precision! Also, don't write $12$ for ${1,2}$ etc.
$endgroup$
– Henno Brandsma
Feb 1 at 23:00
$begingroup$
$tau_1$ should go? You still haven't defined $X$. Maths is precision! Also, don't write $12$ for ${1,2}$ etc.
$endgroup$
– Henno Brandsma
Feb 1 at 23:00
add a comment |
2 Answers
2
active
oldest
votes
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$f(1)=1=f(2)$ and $f(3)=f(4)=2$ is not constant.
$f^{-1}[emptyset]=emptysetin tau$ (we can actually omit it, and also
$f^{-1}[X]=X in tau$ as these hold for any function between two sets),
$f^{-1}[{1}]={1,2} in tau$
$f^{-1}[{1,2}=f^{-1}[{1,2,3}]=Xin tau$. So $f$ is continuous.
answered Feb 1 at 23:09
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
add a comment |
$begingroup$
Exactly. In particular we need $f^{-1}({1})intau$, so it can be either ${1,2}$ or ${3,4}$ or $emptyset$ (we exclude $X$ since $f$ ought not to be constant).
The first two options are symmetric, and you can continue this process (with either choice) to construct a continuous $f$..
answered Feb 1 at 22:59
BerciBerci
62k23776
$endgroup$
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$f(1)=1=f(2)$ and $f(3)=f(4)=2$ is not constant.
$f^{-1}[emptyset]=emptysetin tau$ (we can actually omit it, and also
$f^{-1}[X]=X in tau$ as these hold for any function between two sets),
$f^{-1}[{1}]={1,2} in tau$
$f^{-1}[{1,2}=f^{-1}[{1,2,3}]=Xin tau$. So $f$ is continuous.
answered Feb 1 at 23:09
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
add a comment |
$begingroup$
$f(1)=1=f(2)$ and $f(3)=f(4)=2$ is not constant.
$f^{-1}[emptyset]=emptysetin tau$ (we can actually omit it, and also
$f^{-1}[X]=X in tau$ as these hold for any function between two sets),
$f^{-1}[{1}]={1,2} in tau$
$f^{-1}[{1,2}=f^{-1}[{1,2,3}]=Xin tau$. So $f$ is continuous.
answered Feb 1 at 23:09
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
add a comment |
$begingroup$
$f(1)=1=f(2)$ and $f(3)=f(4)=2$ is not constant.
$f^{-1}[emptyset]=emptysetin tau$ (we can actually omit it, and also
$f^{-1}[X]=X in tau$ as these hold for any function between two sets),
$f^{-1}[{1}]={1,2} in tau$
$f^{-1}[{1,2}=f^{-1}[{1,2,3}]=Xin tau$. So $f$ is continuous.
answered Feb 1 at 23:09
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
$f(1)=1=f(2)$ and $f(3)=f(4)=2$ is not constant.
$f^{-1}[emptyset]=emptysetin tau$ (we can actually omit it, and also
$f^{-1}[X]=X in tau$ as these hold for any function between two sets),
$f^{-1}[{1}]={1,2} in tau$
$f^{-1}[{1,2}=f^{-1}[{1,2,3}]=Xin tau$. So $f$ is continuous.
answered Feb 1 at 23:09
Henno BrandsmaHenno Brandsma
116k349127
edited Feb 2 at 9:15
answered Feb 1 at 23:09
Henno BrandsmaHenno Brandsma
116k349127
answered Feb 1 at 23:09
Henno BrandsmaHenno Brandsma
116k349127
answered Feb 1 at 23:09
Henno BrandsmaHenno Brandsma
116k349127
116k349127
add a comment |
add a comment |
$begingroup$
Exactly. In particular we need $f^{-1}({1})intau$, so it can be either ${1,2}$ or ${3,4}$ or $emptyset$ (we exclude $X$ since $f$ ought not to be constant).
The first two options are symmetric, and you can continue this process (with either choice) to construct a continuous $f$..
answered Feb 1 at 22:59
BerciBerci
62k23776
$endgroup$
add a comment |
$begingroup$
Exactly. In particular we need $f^{-1}({1})intau$, so it can be either ${1,2}$ or ${3,4}$ or $emptyset$ (we exclude $X$ since $f$ ought not to be constant).
The first two options are symmetric, and you can continue this process (with either choice) to construct a continuous $f$..
answered Feb 1 at 22:59
BerciBerci
62k23776
$endgroup$
add a comment |
$begingroup$
Exactly. In particular we need $f^{-1}({1})intau$, so it can be either ${1,2}$ or ${3,4}$ or $emptyset$ (we exclude $X$ since $f$ ought not to be constant).
The first two options are symmetric, and you can continue this process (with either choice) to construct a continuous $f$..
answered Feb 1 at 22:59
BerciBerci
62k23776
$endgroup$
Exactly. In particular we need $f^{-1}({1})intau$, so it can be either ${1,2}$ or ${3,4}$ or $emptyset$ (we exclude $X$ since $f$ ought not to be constant).
The first two options are symmetric, and you can continue this process (with either choice) to construct a continuous $f$..
answered Feb 1 at 22:59
BerciBerci
62k23776
answered Feb 1 at 22:59
BerciBerci
62k23776
answered Feb 1 at 22:59
BerciBerci
62k23776
answered Feb 1 at 22:59
BerciBerci
62k23776
62k23776
add a comment |
add a comment |
1
$begingroup$
What is $tau_1$? From what space to what space does $f$ have to go? The question is unclear. $X={1,2,3,4}$?
$endgroup$
– Henno Brandsma
Feb 1 at 22:56
$begingroup$
@HennoBrandsma Please check my edit.
$endgroup$
– Pedro Gomes
Feb 1 at 22:59
1
$begingroup$
$tau_1$ should go? You still haven't defined $X$. Maths is precision! Also, don't write $12$ for ${1,2}$ etc.
$endgroup$
– Henno Brandsma
Feb 1 at 23:00