Show that $tau$ is a topology on $mathbb{R}$ [closed]
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I really wanted to solve this math problem on my own, but I have absolutely no idea how to attack this exercise and REALLY needs some hints:
Let $tau$ be the system of subsets U in $mathbb{R}$ which is one of the following types:
Either:
(i) U does not contain $0$,
(ii) U does contain $0$, and the complementary set $mathbb{R}$ U is finite.
SHOW that $tau$ is a topology on $mathbb{R}$.
general-topology topological-groups
asked Jan 10 at 23:44
MathStudMathStud
83
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closed as off-topic by José Carlos Santos, Shailesh, mrtaurho, Jean-Claude Arbaut, KReiser Jan 11 at 9:03
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." – José Carlos Santos, Shailesh, mrtaurho, Jean-Claude Arbaut, KReiser
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
I really wanted to solve this math problem on my own, but I have absolutely no idea how to attack this exercise and REALLY needs some hints:
Let $tau$ be the system of subsets U in $mathbb{R}$ which is one of the following types:
Either:
(i) U does not contain $0$,
(ii) U does contain $0$, and the complementary set $mathbb{R}$ U is finite.
SHOW that $tau$ is a topology on $mathbb{R}$.
general-topology topological-groups
asked Jan 10 at 23:44
MathStudMathStud
83
$endgroup$
closed as off-topic by José Carlos Santos, Shailesh, mrtaurho, Jean-Claude Arbaut, KReiser Jan 11 at 9:03
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." – José Carlos Santos, Shailesh, mrtaurho, Jean-Claude Arbaut, KReiser
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
I noticed you originally attempted to use Tau in the title. Capital $tau$ is just a $T$ for this purpose.
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– Rhys Hughes
Jan 10 at 23:47
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what have you tried? To start, what are the axioms of a topology on a set $X$?
$endgroup$
– Juan Diego Rojas
Jan 10 at 23:52
$begingroup$
To show $tau$ is a topology, you need to show it contains the empty set and the entire set, and that it is closed under arbitrary unions and finite intersections. Take a family of sets $(U_{lambda}|lambda in Lambda)$ in $tau$ for some indexing set $Lambda$. What can you say about their union? What can you say about the intersection of finitely many $U_{lambda}$?
$endgroup$
– greelious
Jan 10 at 23:53
add a comment |
$begingroup$
I really wanted to solve this math problem on my own, but I have absolutely no idea how to attack this exercise and REALLY needs some hints:
Let $tau$ be the system of subsets U in $mathbb{R}$ which is one of the following types:
Either:
(i) U does not contain $0$,
(ii) U does contain $0$, and the complementary set $mathbb{R}$ U is finite.
SHOW that $tau$ is a topology on $mathbb{R}$.
general-topology topological-groups
asked Jan 10 at 23:44
MathStudMathStud
83
$endgroup$
I really wanted to solve this math problem on my own, but I have absolutely no idea how to attack this exercise and REALLY needs some hints:
Let $tau$ be the system of subsets U in $mathbb{R}$ which is one of the following types:
Either:
(i) U does not contain $0$,
(ii) U does contain $0$, and the complementary set $mathbb{R}$ U is finite.
SHOW that $tau$ is a topology on $mathbb{R}$.
general-topology topological-groups
general-topology topological-groups
asked Jan 10 at 23:44
MathStudMathStud
83
asked Jan 10 at 23:44
MathStudMathStud
83
asked Jan 10 at 23:44
MathStudMathStud
83
asked Jan 10 at 23:44
MathStudMathStud
83
asked Jan 10 at 23:44
MathStudMathStud
83
83
closed as off-topic by José Carlos Santos, Shailesh, mrtaurho, Jean-Claude Arbaut, KReiser Jan 11 at 9:03
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." – José Carlos Santos, Shailesh, mrtaurho, Jean-Claude Arbaut, KReiser
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by José Carlos Santos, Shailesh, mrtaurho, Jean-Claude Arbaut, KReiser Jan 11 at 9:03
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." – José Carlos Santos, Shailesh, mrtaurho, Jean-Claude Arbaut, KReiser
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
I noticed you originally attempted to use Tau in the title. Capital $tau$ is just a $T$ for this purpose.
$endgroup$
– Rhys Hughes
Jan 10 at 23:47
$begingroup$
what have you tried? To start, what are the axioms of a topology on a set $X$?
$endgroup$
– Juan Diego Rojas
Jan 10 at 23:52
$begingroup$
To show $tau$ is a topology, you need to show it contains the empty set and the entire set, and that it is closed under arbitrary unions and finite intersections. Take a family of sets $(U_{lambda}|lambda in Lambda)$ in $tau$ for some indexing set $Lambda$. What can you say about their union? What can you say about the intersection of finitely many $U_{lambda}$?
$endgroup$
– greelious
Jan 10 at 23:53
add a comment |
$begingroup$
I noticed you originally attempted to use Tau in the title. Capital $tau$ is just a $T$ for this purpose.
$endgroup$
– Rhys Hughes
Jan 10 at 23:47
$begingroup$
what have you tried? To start, what are the axioms of a topology on a set $X$?
$endgroup$
– Juan Diego Rojas
Jan 10 at 23:52
$begingroup$
To show $tau$ is a topology, you need to show it contains the empty set and the entire set, and that it is closed under arbitrary unions and finite intersections. Take a family of sets $(U_{lambda}|lambda in Lambda)$ in $tau$ for some indexing set $Lambda$. What can you say about their union? What can you say about the intersection of finitely many $U_{lambda}$?
$endgroup$
– greelious
Jan 10 at 23:53
$begingroup$
I noticed you originally attempted to use Tau in the title. Capital $tau$ is just a $T$ for this purpose.
$endgroup$
– Rhys Hughes
Jan 10 at 23:47
$begingroup$
I noticed you originally attempted to use Tau in the title. Capital $tau$ is just a $T$ for this purpose.
$endgroup$
– Rhys Hughes
Jan 10 at 23:47
$begingroup$
what have you tried? To start, what are the axioms of a topology on a set $X$?
$endgroup$
– Juan Diego Rojas
Jan 10 at 23:52
$begingroup$
what have you tried? To start, what are the axioms of a topology on a set $X$?
$endgroup$
– Juan Diego Rojas
Jan 10 at 23:52
$begingroup$
To show $tau$ is a topology, you need to show it contains the empty set and the entire set, and that it is closed under arbitrary unions and finite intersections. Take a family of sets $(U_{lambda}|lambda in Lambda)$ in $tau$ for some indexing set $Lambda$. What can you say about their union? What can you say about the intersection of finitely many $U_{lambda}$?
$endgroup$
– greelious
Jan 10 at 23:53
$begingroup$
To show $tau$ is a topology, you need to show it contains the empty set and the entire set, and that it is closed under arbitrary unions and finite intersections. Take a family of sets $(U_{lambda}|lambda in Lambda)$ in $tau$ for some indexing set $Lambda$. What can you say about their union? What can you say about the intersection of finitely many $U_{lambda}$?
$endgroup$
– greelious
Jan 10 at 23:53
add a comment |
1 Answer
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All you need to do is check that $$tau = { U subseteq mathbb{R} | U text{ doesn't contain $0$ or U contains $0$ and $mathbb{R} setminus U$ is finite}} $$ actually satisfies the axioms for a topology on $mathbb{R}$.
So you need to do the following
- Show that $mathbb{R}$ and $emptyset$ are elements of $tau$
- Choose any collection of elements ${U_i}_{i in I}$ of $tau$ (i.e $U_i in tau$ for each $i in I$) and show that $bigcup_{i in I} U_i in tau$
- Choose any finite collection of elements of $tau$, ${V_1, dots, V_n}$ and show that $bigcap_{i=1}^n V_i in tau$
Then $tau$ is a topology on $mathbb{R}$, by the definiton of a topology on a set.
As an example I'll check one small part of the above for you. I'll show that $mathbb{R} in tau$. Note that $mathbb{R}$ contains $0$ and $mathbb{R} setminus mathbb{R} = emptyset$ which is certainly finite, and so $mathbb{R} in tau$.
answered Jan 10 at 23:54
PerturbativePerturbative
4,41621553
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add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
All you need to do is check that $$tau = { U subseteq mathbb{R} | U text{ doesn't contain $0$ or U contains $0$ and $mathbb{R} setminus U$ is finite}} $$ actually satisfies the axioms for a topology on $mathbb{R}$.
So you need to do the following
- Show that $mathbb{R}$ and $emptyset$ are elements of $tau$
- Choose any collection of elements ${U_i}_{i in I}$ of $tau$ (i.e $U_i in tau$ for each $i in I$) and show that $bigcup_{i in I} U_i in tau$
- Choose any finite collection of elements of $tau$, ${V_1, dots, V_n}$ and show that $bigcap_{i=1}^n V_i in tau$
Then $tau$ is a topology on $mathbb{R}$, by the definiton of a topology on a set.
As an example I'll check one small part of the above for you. I'll show that $mathbb{R} in tau$. Note that $mathbb{R}$ contains $0$ and $mathbb{R} setminus mathbb{R} = emptyset$ which is certainly finite, and so $mathbb{R} in tau$.
answered Jan 10 at 23:54
PerturbativePerturbative
4,41621553
$endgroup$
add a comment |
$begingroup$
All you need to do is check that $$tau = { U subseteq mathbb{R} | U text{ doesn't contain $0$ or U contains $0$ and $mathbb{R} setminus U$ is finite}} $$ actually satisfies the axioms for a topology on $mathbb{R}$.
So you need to do the following
- Show that $mathbb{R}$ and $emptyset$ are elements of $tau$
- Choose any collection of elements ${U_i}_{i in I}$ of $tau$ (i.e $U_i in tau$ for each $i in I$) and show that $bigcup_{i in I} U_i in tau$
- Choose any finite collection of elements of $tau$, ${V_1, dots, V_n}$ and show that $bigcap_{i=1}^n V_i in tau$
Then $tau$ is a topology on $mathbb{R}$, by the definiton of a topology on a set.
As an example I'll check one small part of the above for you. I'll show that $mathbb{R} in tau$. Note that $mathbb{R}$ contains $0$ and $mathbb{R} setminus mathbb{R} = emptyset$ which is certainly finite, and so $mathbb{R} in tau$.
answered Jan 10 at 23:54
PerturbativePerturbative
4,41621553
$endgroup$
add a comment |
$begingroup$
All you need to do is check that $$tau = { U subseteq mathbb{R} | U text{ doesn't contain $0$ or U contains $0$ and $mathbb{R} setminus U$ is finite}} $$ actually satisfies the axioms for a topology on $mathbb{R}$.
So you need to do the following
- Show that $mathbb{R}$ and $emptyset$ are elements of $tau$
- Choose any collection of elements ${U_i}_{i in I}$ of $tau$ (i.e $U_i in tau$ for each $i in I$) and show that $bigcup_{i in I} U_i in tau$
- Choose any finite collection of elements of $tau$, ${V_1, dots, V_n}$ and show that $bigcap_{i=1}^n V_i in tau$
Then $tau$ is a topology on $mathbb{R}$, by the definiton of a topology on a set.
As an example I'll check one small part of the above for you. I'll show that $mathbb{R} in tau$. Note that $mathbb{R}$ contains $0$ and $mathbb{R} setminus mathbb{R} = emptyset$ which is certainly finite, and so $mathbb{R} in tau$.
answered Jan 10 at 23:54
PerturbativePerturbative
4,41621553
$endgroup$
All you need to do is check that $$tau = { U subseteq mathbb{R} | U text{ doesn't contain $0$ or U contains $0$ and $mathbb{R} setminus U$ is finite}} $$ actually satisfies the axioms for a topology on $mathbb{R}$.
So you need to do the following
- Show that $mathbb{R}$ and $emptyset$ are elements of $tau$
- Choose any collection of elements ${U_i}_{i in I}$ of $tau$ (i.e $U_i in tau$ for each $i in I$) and show that $bigcup_{i in I} U_i in tau$
- Choose any finite collection of elements of $tau$, ${V_1, dots, V_n}$ and show that $bigcap_{i=1}^n V_i in tau$
Then $tau$ is a topology on $mathbb{R}$, by the definiton of a topology on a set.
As an example I'll check one small part of the above for you. I'll show that $mathbb{R} in tau$. Note that $mathbb{R}$ contains $0$ and $mathbb{R} setminus mathbb{R} = emptyset$ which is certainly finite, and so $mathbb{R} in tau$.
answered Jan 10 at 23:54
PerturbativePerturbative
4,41621553
answered Jan 10 at 23:54
PerturbativePerturbative
4,41621553
answered Jan 10 at 23:54
PerturbativePerturbative
4,41621553
answered Jan 10 at 23:54
PerturbativePerturbative
4,41621553
4,41621553
add a comment |
add a comment |
$begingroup$
I noticed you originally attempted to use Tau in the title. Capital $tau$ is just a $T$ for this purpose.
$endgroup$
– Rhys Hughes
Jan 10 at 23:47
$begingroup$
what have you tried? To start, what are the axioms of a topology on a set $X$?
$endgroup$
– Juan Diego Rojas
Jan 10 at 23:52
$begingroup$
To show $tau$ is a topology, you need to show it contains the empty set and the entire set, and that it is closed under arbitrary unions and finite intersections. Take a family of sets $(U_{lambda}|lambda in Lambda)$ in $tau$ for some indexing set $Lambda$. What can you say about their union? What can you say about the intersection of finitely many $U_{lambda}$?
$endgroup$
– greelious
Jan 10 at 23:53