Separable metric topology refined with an $F_sigma$-set [closed]
0
$begingroup$
Let $(X,tau)$ be a separable metrizable space. Let $A$ be an $F_sigma$-subset of $X$. Is the topology generated by $taucup {A}$ also metrizable?
general-topology
asked Dec 15 '18 at 22:39
aposyndeticaposyndetic
1
$endgroup$
closed as off-topic by Shailesh, Paul Frost, Eevee Trainer, Saad, Cesareo Dec 16 '18 at 1:18
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Shailesh, Paul Frost, Eevee Trainer, Saad, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
0
$begingroup$
Let $(X,tau)$ be a separable metrizable space. Let $A$ be an $F_sigma$-subset of $X$. Is the topology generated by $taucup {A}$ also metrizable?
general-topology
asked Dec 15 '18 at 22:39
aposyndeticaposyndetic
1
$endgroup$
closed as off-topic by Shailesh, Paul Frost, Eevee Trainer, Saad, Cesareo Dec 16 '18 at 1:18
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Shailesh, Paul Frost, Eevee Trainer, Saad, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
Any thoughts on the question you'd like to add to your post? Or, could you add the source of the question, and your motivation for asking it? Could you specify, in your post, what exactly you are unsure of? Care to share the definition of metrizable space you are working with? (Definitions are your friend, in questions like this.)
$endgroup$
– amWhy
Dec 15 '18 at 22:44
add a comment |
0
0
0
$begingroup$
Let $(X,tau)$ be a separable metrizable space. Let $A$ be an $F_sigma$-subset of $X$. Is the topology generated by $taucup {A}$ also metrizable?
general-topology
asked Dec 15 '18 at 22:39
aposyndeticaposyndetic
1
$endgroup$
Let $(X,tau)$ be a separable metrizable space. Let $A$ be an $F_sigma$-subset of $X$. Is the topology generated by $taucup {A}$ also metrizable?
general-topology
general-topology
asked Dec 15 '18 at 22:39
aposyndeticaposyndetic
1
asked Dec 15 '18 at 22:39
aposyndeticaposyndetic
1
asked Dec 15 '18 at 22:39
aposyndeticaposyndetic
1
asked Dec 15 '18 at 22:39
aposyndeticaposyndetic
1
asked Dec 15 '18 at 22:39
aposyndeticaposyndetic
1
1
closed as off-topic by Shailesh, Paul Frost, Eevee Trainer, Saad, Cesareo Dec 16 '18 at 1:18
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Shailesh, Paul Frost, Eevee Trainer, Saad, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Shailesh, Paul Frost, Eevee Trainer, Saad, Cesareo Dec 16 '18 at 1:18
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Shailesh, Paul Frost, Eevee Trainer, Saad, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
Any thoughts on the question you'd like to add to your post? Or, could you add the source of the question, and your motivation for asking it? Could you specify, in your post, what exactly you are unsure of? Care to share the definition of metrizable space you are working with? (Definitions are your friend, in questions like this.)
$endgroup$
– amWhy
Dec 15 '18 at 22:44
add a comment |
$begingroup$
Any thoughts on the question you'd like to add to your post? Or, could you add the source of the question, and your motivation for asking it? Could you specify, in your post, what exactly you are unsure of? Care to share the definition of metrizable space you are working with? (Definitions are your friend, in questions like this.)
$endgroup$
– amWhy
Dec 15 '18 at 22:44
$begingroup$
Any thoughts on the question you'd like to add to your post? Or, could you add the source of the question, and your motivation for asking it? Could you specify, in your post, what exactly you are unsure of? Care to share the definition of metrizable space you are working with? (Definitions are your friend, in questions like this.)
$endgroup$
– amWhy
Dec 15 '18 at 22:44
$begingroup$
Any thoughts on the question you'd like to add to your post? Or, could you add the source of the question, and your motivation for asking it? Could you specify, in your post, what exactly you are unsure of? Care to share the definition of metrizable space you are working with? (Definitions are your friend, in questions like this.)
$endgroup$
– amWhy
Dec 15 '18 at 22:44
add a comment |
1 Answer
1
active
oldest
votes
2
$begingroup$
No, take $A$ to be the rationals and $tau$ the standard topology on the reals. It's a standard fact (recall Munkres $K$-topology e.g.) that the topology generated by $tau cup {A}$ is not even regular, let alone metrisable.
answered Dec 15 '18 at 22:48
Henno BrandsmaHenno Brandsma
106k347114
$endgroup$
$begingroup$
$tau$ is the standard topology on the reals?
$endgroup$
– aposyndetic
Dec 15 '18 at 22:53
$begingroup$
@aposyndetic yes, I clarified.
$endgroup$
– Henno Brandsma
Dec 15 '18 at 22:54
$begingroup$
also what if that topology was regular? Then wouldn't it be metrizable then?
$endgroup$
– aposyndetic
Dec 15 '18 at 22:55
$begingroup$
@aposyndetic For the reals, yes: we'd have a second countable regular space (so a metrisable one). Regularity is the main obstacle.
$endgroup$
– Henno Brandsma
Dec 15 '18 at 22:59
$begingroup$
so for instance a space which is a countable union of compact metric spaces would be metrizable?
$endgroup$
– aposyndetic
Dec 15 '18 at 23:00
|
show 2 more comments
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
2
$begingroup$
No, take $A$ to be the rationals and $tau$ the standard topology on the reals. It's a standard fact (recall Munkres $K$-topology e.g.) that the topology generated by $tau cup {A}$ is not even regular, let alone metrisable.
answered Dec 15 '18 at 22:48
Henno BrandsmaHenno Brandsma
106k347114
$endgroup$
$begingroup$
$tau$ is the standard topology on the reals?
$endgroup$
– aposyndetic
Dec 15 '18 at 22:53
$begingroup$
@aposyndetic yes, I clarified.
$endgroup$
– Henno Brandsma
Dec 15 '18 at 22:54
$begingroup$
also what if that topology was regular? Then wouldn't it be metrizable then?
$endgroup$
– aposyndetic
Dec 15 '18 at 22:55
$begingroup$
@aposyndetic For the reals, yes: we'd have a second countable regular space (so a metrisable one). Regularity is the main obstacle.
$endgroup$
– Henno Brandsma
Dec 15 '18 at 22:59
$begingroup$
so for instance a space which is a countable union of compact metric spaces would be metrizable?
$endgroup$
– aposyndetic
Dec 15 '18 at 23:00
|
show 2 more comments
2
$begingroup$
No, take $A$ to be the rationals and $tau$ the standard topology on the reals. It's a standard fact (recall Munkres $K$-topology e.g.) that the topology generated by $tau cup {A}$ is not even regular, let alone metrisable.
answered Dec 15 '18 at 22:48
Henno BrandsmaHenno Brandsma
106k347114
$endgroup$
$begingroup$
$tau$ is the standard topology on the reals?
$endgroup$
– aposyndetic
Dec 15 '18 at 22:53
$begingroup$
@aposyndetic yes, I clarified.
$endgroup$
– Henno Brandsma
Dec 15 '18 at 22:54
$begingroup$
also what if that topology was regular? Then wouldn't it be metrizable then?
$endgroup$
– aposyndetic
Dec 15 '18 at 22:55
$begingroup$
@aposyndetic For the reals, yes: we'd have a second countable regular space (so a metrisable one). Regularity is the main obstacle.
$endgroup$
– Henno Brandsma
Dec 15 '18 at 22:59
$begingroup$
so for instance a space which is a countable union of compact metric spaces would be metrizable?
$endgroup$
– aposyndetic
Dec 15 '18 at 23:00
|
show 2 more comments
2
2
2
$begingroup$
No, take $A$ to be the rationals and $tau$ the standard topology on the reals. It's a standard fact (recall Munkres $K$-topology e.g.) that the topology generated by $tau cup {A}$ is not even regular, let alone metrisable.
answered Dec 15 '18 at 22:48
Henno BrandsmaHenno Brandsma
106k347114
$endgroup$
No, take $A$ to be the rationals and $tau$ the standard topology on the reals. It's a standard fact (recall Munkres $K$-topology e.g.) that the topology generated by $tau cup {A}$ is not even regular, let alone metrisable.
answered Dec 15 '18 at 22:48
Henno BrandsmaHenno Brandsma
106k347114
edited Dec 16 '18 at 7:13
answered Dec 15 '18 at 22:48
Henno BrandsmaHenno Brandsma
106k347114
answered Dec 15 '18 at 22:48
Henno BrandsmaHenno Brandsma
106k347114
answered Dec 15 '18 at 22:48
Henno BrandsmaHenno Brandsma
106k347114
106k347114
$begingroup$
$tau$ is the standard topology on the reals?
$endgroup$
– aposyndetic
Dec 15 '18 at 22:53
$begingroup$
@aposyndetic yes, I clarified.
$endgroup$
– Henno Brandsma
Dec 15 '18 at 22:54
$begingroup$
also what if that topology was regular? Then wouldn't it be metrizable then?
$endgroup$
– aposyndetic
Dec 15 '18 at 22:55
$begingroup$
@aposyndetic For the reals, yes: we'd have a second countable regular space (so a metrisable one). Regularity is the main obstacle.
$endgroup$
– Henno Brandsma
Dec 15 '18 at 22:59
$begingroup$
so for instance a space which is a countable union of compact metric spaces would be metrizable?
$endgroup$
– aposyndetic
Dec 15 '18 at 23:00
|
show 2 more comments
$begingroup$
$tau$ is the standard topology on the reals?
$endgroup$
– aposyndetic
Dec 15 '18 at 22:53
$begingroup$
@aposyndetic yes, I clarified.
$endgroup$
– Henno Brandsma
Dec 15 '18 at 22:54
$begingroup$
also what if that topology was regular? Then wouldn't it be metrizable then?
$endgroup$
– aposyndetic
Dec 15 '18 at 22:55
$begingroup$
@aposyndetic For the reals, yes: we'd have a second countable regular space (so a metrisable one). Regularity is the main obstacle.
$endgroup$
– Henno Brandsma
Dec 15 '18 at 22:59
$begingroup$
so for instance a space which is a countable union of compact metric spaces would be metrizable?
$endgroup$
– aposyndetic
Dec 15 '18 at 23:00
$begingroup$
$tau$ is the standard topology on the reals?
$endgroup$
– aposyndetic
Dec 15 '18 at 22:53
$begingroup$
$tau$ is the standard topology on the reals?
$endgroup$
– aposyndetic
Dec 15 '18 at 22:53
$begingroup$
@aposyndetic yes, I clarified.
$endgroup$
– Henno Brandsma
Dec 15 '18 at 22:54
$begingroup$
@aposyndetic yes, I clarified.
$endgroup$
– Henno Brandsma
Dec 15 '18 at 22:54
$begingroup$
also what if that topology was regular? Then wouldn't it be metrizable then?
$endgroup$
– aposyndetic
Dec 15 '18 at 22:55
$begingroup$
also what if that topology was regular? Then wouldn't it be metrizable then?
$endgroup$
– aposyndetic
Dec 15 '18 at 22:55
$begingroup$
@aposyndetic For the reals, yes: we'd have a second countable regular space (so a metrisable one). Regularity is the main obstacle.
$endgroup$
– Henno Brandsma
Dec 15 '18 at 22:59
$begingroup$
@aposyndetic For the reals, yes: we'd have a second countable regular space (so a metrisable one). Regularity is the main obstacle.
$endgroup$
– Henno Brandsma
Dec 15 '18 at 22:59
$begingroup$
so for instance a space which is a countable union of compact metric spaces would be metrizable?
$endgroup$
– aposyndetic
Dec 15 '18 at 23:00
$begingroup$
so for instance a space which is a countable union of compact metric spaces would be metrizable?
$endgroup$
– aposyndetic
Dec 15 '18 at 23:00
|
show 2 more comments
$begingroup$
Any thoughts on the question you'd like to add to your post? Or, could you add the source of the question, and your motivation for asking it? Could you specify, in your post, what exactly you are unsure of? Care to share the definition of metrizable space you are working with? (Definitions are your friend, in questions like this.)
$endgroup$
– amWhy
Dec 15 '18 at 22:44