How to find the dense subsets of a topological space [closed]
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I have the following problem and I'm having trouble putting the theory in practice.
Let $X = a,b,c,d $ with the topology $T = emptyset, a, b, a,b, X$. Find the subsets of $X$ that are dense in $X$.
Also, if you can help me with some theory for newbies in this subject, it would be awesome. :3
general-topology
closed as off-topic by uniquesolution, José Carlos Santos, Jendrik Stelzner, A. Goodier, Theoretical Economist Aug 29 at 12:54
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 improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." â uniquesolution, José Carlos Santos, Jendrik Stelzner, A. Goodier, Theoretical Economist
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up vote
-2
down vote
favorite
I have the following problem and I'm having trouble putting the theory in practice.
Let $X = a,b,c,d $ with the topology $T = emptyset, a, b, a,b, X$. Find the subsets of $X$ that are dense in $X$.
Also, if you can help me with some theory for newbies in this subject, it would be awesome. :3
general-topology
closed as off-topic by uniquesolution, José Carlos Santos, Jendrik Stelzner, A. Goodier, Theoretical Economist Aug 29 at 12:54
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 improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." â uniquesolution, José Carlos Santos, Jendrik Stelzner, A. Goodier, Theoretical Economist
Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
â José Carlos Santos
Aug 29 at 7:13
I have the theory upfront, but i really can't understand the concepts. I've seen this post math.stackexchange.com/questions/1797859/⦠and i get the fact that if i take let's say M = b â T then the dense subsets are a,b, b,c and b,d. Is it correct?
â Timur Mengazi
Aug 29 at 7:20
That doesn't make sense.
â José Carlos Santos
Aug 29 at 7:30
I know it doesn't that's why I'm here, I'm trying to understand it...
â Timur Mengazi
Aug 29 at 7:34
add a comment |Â
up vote
-2
down vote
favorite
up vote
-2
down vote
favorite
I have the following problem and I'm having trouble putting the theory in practice.
Let $X = a,b,c,d $ with the topology $T = emptyset, a, b, a,b, X$. Find the subsets of $X$ that are dense in $X$.
Also, if you can help me with some theory for newbies in this subject, it would be awesome. :3
general-topology
I have the following problem and I'm having trouble putting the theory in practice.
Let $X = a,b,c,d $ with the topology $T = emptyset, a, b, a,b, X$. Find the subsets of $X$ that are dense in $X$.
Also, if you can help me with some theory for newbies in this subject, it would be awesome. :3
general-topology
edited Aug 29 at 11:15
Jendrik Stelzner
7,69121037
7,69121037
asked Aug 29 at 7:07
Timur Mengazi
83
83
closed as off-topic by uniquesolution, José Carlos Santos, Jendrik Stelzner, A. Goodier, Theoretical Economist Aug 29 at 12:54
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 improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." â uniquesolution, José Carlos Santos, Jendrik Stelzner, A. Goodier, Theoretical Economist
closed as off-topic by uniquesolution, José Carlos Santos, Jendrik Stelzner, A. Goodier, Theoretical Economist Aug 29 at 12:54
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 improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." â uniquesolution, José Carlos Santos, Jendrik Stelzner, A. Goodier, Theoretical Economist
Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
â José Carlos Santos
Aug 29 at 7:13
I have the theory upfront, but i really can't understand the concepts. I've seen this post math.stackexchange.com/questions/1797859/⦠and i get the fact that if i take let's say M = b â T then the dense subsets are a,b, b,c and b,d. Is it correct?
â Timur Mengazi
Aug 29 at 7:20
That doesn't make sense.
â José Carlos Santos
Aug 29 at 7:30
I know it doesn't that's why I'm here, I'm trying to understand it...
â Timur Mengazi
Aug 29 at 7:34
add a comment |Â
Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
â José Carlos Santos
Aug 29 at 7:13
I have the theory upfront, but i really can't understand the concepts. I've seen this post math.stackexchange.com/questions/1797859/⦠and i get the fact that if i take let's say M = b â T then the dense subsets are a,b, b,c and b,d. Is it correct?
â Timur Mengazi
Aug 29 at 7:20
That doesn't make sense.
â José Carlos Santos
Aug 29 at 7:30
I know it doesn't that's why I'm here, I'm trying to understand it...
â Timur Mengazi
Aug 29 at 7:34
Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
â José Carlos Santos
Aug 29 at 7:13
Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
â José Carlos Santos
Aug 29 at 7:13
I have the theory upfront, but i really can't understand the concepts. I've seen this post math.stackexchange.com/questions/1797859/⦠and i get the fact that if i take let's say M = b â T then the dense subsets are a,b, b,c and b,d. Is it correct?
â Timur Mengazi
Aug 29 at 7:20
I have the theory upfront, but i really can't understand the concepts. I've seen this post math.stackexchange.com/questions/1797859/⦠and i get the fact that if i take let's say M = b â T then the dense subsets are a,b, b,c and b,d. Is it correct?
â Timur Mengazi
Aug 29 at 7:20
That doesn't make sense.
â José Carlos Santos
Aug 29 at 7:30
That doesn't make sense.
â José Carlos Santos
Aug 29 at 7:30
I know it doesn't that's why I'm here, I'm trying to understand it...
â Timur Mengazi
Aug 29 at 7:34
I know it doesn't that's why I'm here, I'm trying to understand it...
â Timur Mengazi
Aug 29 at 7:34
add a comment |Â
1 Answer
1
active
oldest
votes
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0
down vote
accepted
Take $A=a,b$. Then $A$ is dense in $X$ as any non empty open set intersects $A$. Also you can take another dense set say $a,b,c$...this is also a dense set since any non empty open set intersects it...In this way u can find out all the dense set in this space.
1
And you cannot have any other set be dense, because $a$ is a nonempty open set in your topology, so any dense set needs a nonempty intersection with $a$, which means it must contain $a$, the same goes for $b$. So any dense set must be a superset of $a,b$, as stated above.
â Ingix
Aug 29 at 7:30
Let me try to understand what you just said...when you're referring to "non empty open set" you're referring to any set of X with at least 2 elements?
â Timur Mengazi
Aug 29 at 7:32
@TimurMengazi....NO..non empty means "having at least one element"...for example in this case the non empty open sets are precisely $a$, $b$, $a,b$ and $X$
â Indrajit Ghosh
Aug 29 at 7:34
so if I had T = O, b, c , b,c, X then the dense subsets are b,c and b,c,d and a,b,c, a,b,c,d bcs they have at least one element when intersected with T?
â Timur Mengazi
Aug 29 at 7:41
@TimurMengazi...Yes..and same logic
â Indrajit Ghosh
Aug 29 at 7:42
 |Â
show 1 more comment
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
Take $A=a,b$. Then $A$ is dense in $X$ as any non empty open set intersects $A$. Also you can take another dense set say $a,b,c$...this is also a dense set since any non empty open set intersects it...In this way u can find out all the dense set in this space.
1
And you cannot have any other set be dense, because $a$ is a nonempty open set in your topology, so any dense set needs a nonempty intersection with $a$, which means it must contain $a$, the same goes for $b$. So any dense set must be a superset of $a,b$, as stated above.
â Ingix
Aug 29 at 7:30
Let me try to understand what you just said...when you're referring to "non empty open set" you're referring to any set of X with at least 2 elements?
â Timur Mengazi
Aug 29 at 7:32
@TimurMengazi....NO..non empty means "having at least one element"...for example in this case the non empty open sets are precisely $a$, $b$, $a,b$ and $X$
â Indrajit Ghosh
Aug 29 at 7:34
so if I had T = O, b, c , b,c, X then the dense subsets are b,c and b,c,d and a,b,c, a,b,c,d bcs they have at least one element when intersected with T?
â Timur Mengazi
Aug 29 at 7:41
@TimurMengazi...Yes..and same logic
â Indrajit Ghosh
Aug 29 at 7:42
 |Â
show 1 more comment
up vote
0
down vote
accepted
Take $A=a,b$. Then $A$ is dense in $X$ as any non empty open set intersects $A$. Also you can take another dense set say $a,b,c$...this is also a dense set since any non empty open set intersects it...In this way u can find out all the dense set in this space.
1
And you cannot have any other set be dense, because $a$ is a nonempty open set in your topology, so any dense set needs a nonempty intersection with $a$, which means it must contain $a$, the same goes for $b$. So any dense set must be a superset of $a,b$, as stated above.
â Ingix
Aug 29 at 7:30
Let me try to understand what you just said...when you're referring to "non empty open set" you're referring to any set of X with at least 2 elements?
â Timur Mengazi
Aug 29 at 7:32
@TimurMengazi....NO..non empty means "having at least one element"...for example in this case the non empty open sets are precisely $a$, $b$, $a,b$ and $X$
â Indrajit Ghosh
Aug 29 at 7:34
so if I had T = O, b, c , b,c, X then the dense subsets are b,c and b,c,d and a,b,c, a,b,c,d bcs they have at least one element when intersected with T?
â Timur Mengazi
Aug 29 at 7:41
@TimurMengazi...Yes..and same logic
â Indrajit Ghosh
Aug 29 at 7:42
 |Â
show 1 more comment
up vote
0
down vote
accepted
up vote
0
down vote
accepted
Take $A=a,b$. Then $A$ is dense in $X$ as any non empty open set intersects $A$. Also you can take another dense set say $a,b,c$...this is also a dense set since any non empty open set intersects it...In this way u can find out all the dense set in this space.
Take $A=a,b$. Then $A$ is dense in $X$ as any non empty open set intersects $A$. Also you can take another dense set say $a,b,c$...this is also a dense set since any non empty open set intersects it...In this way u can find out all the dense set in this space.
answered Aug 29 at 7:20
Indrajit Ghosh
849516
849516
1
And you cannot have any other set be dense, because $a$ is a nonempty open set in your topology, so any dense set needs a nonempty intersection with $a$, which means it must contain $a$, the same goes for $b$. So any dense set must be a superset of $a,b$, as stated above.
â Ingix
Aug 29 at 7:30
Let me try to understand what you just said...when you're referring to "non empty open set" you're referring to any set of X with at least 2 elements?
â Timur Mengazi
Aug 29 at 7:32
@TimurMengazi....NO..non empty means "having at least one element"...for example in this case the non empty open sets are precisely $a$, $b$, $a,b$ and $X$
â Indrajit Ghosh
Aug 29 at 7:34
so if I had T = O, b, c , b,c, X then the dense subsets are b,c and b,c,d and a,b,c, a,b,c,d bcs they have at least one element when intersected with T?
â Timur Mengazi
Aug 29 at 7:41
@TimurMengazi...Yes..and same logic
â Indrajit Ghosh
Aug 29 at 7:42
 |Â
show 1 more comment
1
And you cannot have any other set be dense, because $a$ is a nonempty open set in your topology, so any dense set needs a nonempty intersection with $a$, which means it must contain $a$, the same goes for $b$. So any dense set must be a superset of $a,b$, as stated above.
â Ingix
Aug 29 at 7:30
Let me try to understand what you just said...when you're referring to "non empty open set" you're referring to any set of X with at least 2 elements?
â Timur Mengazi
Aug 29 at 7:32
@TimurMengazi....NO..non empty means "having at least one element"...for example in this case the non empty open sets are precisely $a$, $b$, $a,b$ and $X$
â Indrajit Ghosh
Aug 29 at 7:34
so if I had T = O, b, c , b,c, X then the dense subsets are b,c and b,c,d and a,b,c, a,b,c,d bcs they have at least one element when intersected with T?
â Timur Mengazi
Aug 29 at 7:41
@TimurMengazi...Yes..and same logic
â Indrajit Ghosh
Aug 29 at 7:42
1
1
And you cannot have any other set be dense, because $a$ is a nonempty open set in your topology, so any dense set needs a nonempty intersection with $a$, which means it must contain $a$, the same goes for $b$. So any dense set must be a superset of $a,b$, as stated above.
â Ingix
Aug 29 at 7:30
And you cannot have any other set be dense, because $a$ is a nonempty open set in your topology, so any dense set needs a nonempty intersection with $a$, which means it must contain $a$, the same goes for $b$. So any dense set must be a superset of $a,b$, as stated above.
â Ingix
Aug 29 at 7:30
Let me try to understand what you just said...when you're referring to "non empty open set" you're referring to any set of X with at least 2 elements?
â Timur Mengazi
Aug 29 at 7:32
Let me try to understand what you just said...when you're referring to "non empty open set" you're referring to any set of X with at least 2 elements?
â Timur Mengazi
Aug 29 at 7:32
@TimurMengazi....NO..non empty means "having at least one element"...for example in this case the non empty open sets are precisely $a$, $b$, $a,b$ and $X$
â Indrajit Ghosh
Aug 29 at 7:34
@TimurMengazi....NO..non empty means "having at least one element"...for example in this case the non empty open sets are precisely $a$, $b$, $a,b$ and $X$
â Indrajit Ghosh
Aug 29 at 7:34
so if I had T = O, b, c , b,c, X then the dense subsets are b,c and b,c,d and a,b,c, a,b,c,d bcs they have at least one element when intersected with T?
â Timur Mengazi
Aug 29 at 7:41
so if I had T = O, b, c , b,c, X then the dense subsets are b,c and b,c,d and a,b,c, a,b,c,d bcs they have at least one element when intersected with T?
â Timur Mengazi
Aug 29 at 7:41
@TimurMengazi...Yes..and same logic
â Indrajit Ghosh
Aug 29 at 7:42
@TimurMengazi...Yes..and same logic
â Indrajit Ghosh
Aug 29 at 7:42
 |Â
show 1 more comment
Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
â José Carlos Santos
Aug 29 at 7:13
I have the theory upfront, but i really can't understand the concepts. I've seen this post math.stackexchange.com/questions/1797859/⦠and i get the fact that if i take let's say M = b â T then the dense subsets are a,b, b,c and b,d. Is it correct?
â Timur Mengazi
Aug 29 at 7:20
That doesn't make sense.
â José Carlos Santos
Aug 29 at 7:30
I know it doesn't that's why I'm here, I'm trying to understand it...
â Timur Mengazi
Aug 29 at 7:34