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Is R is connected subspace of R2.?
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1st I want to know , is R is a subspace of R2?
Because R can be written as R×0 which is a subset of R2.
connectedness
asked Jul 23 at 16:46
Subhasish Chowdhury
63
add a comment |Â
up vote
1
down vote
favorite
1st I want to know , is R is a subspace of R2?
Because R can be written as R×0 which is a subset of R2.
connectedness
asked Jul 23 at 16:46
Subhasish Chowdhury
63
No, it is not a subspace at all.
– Sean Roberson
Jul 23 at 16:47
It depends on how you wish to think about $mathbbR$.
– Randall
Jul 23 at 16:48
$mathbb R$ is connected. It is homeomorphic to $mathbb R times 0 subset mathbb R^2$. You are done once you decide whether that is what you mean by "subspace".
– GEdgar
Jul 23 at 16:58
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
1st I want to know , is R is a subspace of R2?
Because R can be written as R×0 which is a subset of R2.
connectedness
asked Jul 23 at 16:46
Subhasish Chowdhury
63
1st I want to know , is R is a subspace of R2?
Because R can be written as R×0 which is a subset of R2.
connectedness
asked Jul 23 at 16:46
Subhasish Chowdhury
63
asked Jul 23 at 16:46
Subhasish Chowdhury
63
asked Jul 23 at 16:46
Subhasish Chowdhury
63
asked Jul 23 at 16:46
Subhasish Chowdhury
63
63
No, it is not a subspace at all.
– Sean Roberson
Jul 23 at 16:47
It depends on how you wish to think about $mathbbR$.
– Randall
Jul 23 at 16:48
$mathbb R$ is connected. It is homeomorphic to $mathbb R times 0 subset mathbb R^2$. You are done once you decide whether that is what you mean by "subspace".
– GEdgar
Jul 23 at 16:58
add a comment |Â
No, it is not a subspace at all.
– Sean Roberson
Jul 23 at 16:47
It depends on how you wish to think about $mathbbR$.
– Randall
Jul 23 at 16:48
$mathbb R$ is connected. It is homeomorphic to $mathbb R times 0 subset mathbb R^2$. You are done once you decide whether that is what you mean by "subspace".
– GEdgar
Jul 23 at 16:58
No, it is not a subspace at all.
– Sean Roberson
Jul 23 at 16:47
No, it is not a subspace at all.
– Sean Roberson
Jul 23 at 16:47
It depends on how you wish to think about $mathbbR$.
– Randall
Jul 23 at 16:48
It depends on how you wish to think about $mathbbR$.
– Randall
Jul 23 at 16:48
$mathbb R$ is connected. It is homeomorphic to $mathbb R times 0 subset mathbb R^2$. You are done once you decide whether that is what you mean by "subspace".
– GEdgar
Jul 23 at 16:58
$mathbb R$ is connected. It is homeomorphic to $mathbb R times 0 subset mathbb R^2$. You are done once you decide whether that is what you mean by "subspace".
– GEdgar
Jul 23 at 16:58
add a comment |Â
1 Answer
1
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oldest
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up vote
2
down vote
So, you're asking a question which is a lot deeper than you might think. The short and sweet is that $mathbbR$ is one dimensional, but in $mathbbR^2$ you have 2 dimensions. So, $mathbbR times 0$ is a 1 dimensional subspace of a 2 dimensional space, the fact that it is a subspace of a 2 dimensional space is important information we'd like to keep. To get a better grasp of this concept, I suggest reading the book Flatland by Edwin Abbott Abbott.
Now, in a more technical sense, what you're asking about is something called a morphism. In our case, we would say, in plain english, that $mathbbR times 0$ is basically $mathbbR$. It behaves and acts just like it, however it is not $mathbbR$, it is a subset of $mathbbR^2$ that looks just like $mathbbR$. You would be incorrect, or at least remiss, to say that this is $mathbbR$.
Welcome to the beauty and elegance of Mathematics! :)
answered Jul 23 at 16:57
Blake
33219
$mathbbR times 0$ is 1 dimensional since a basis only has 1 element
– LinAlg
Jul 23 at 17:01
Well, it spans a 1 dimensional subspace of $mathbbR^2$, sure. However, it is still two dimensional because it lives within a two dimensional space. You can't tell me that it's 1 dimensional. How do you write an element of $mathbbR times 0$? Would you agree that it looks like $(x, 0) $?
– Blake
Jul 23 at 17:05
It is not two dimensional. The correct way of saying is it is a 1 dimensional subspace of a 2 dimensional space.
– LinAlg
Jul 23 at 17:06
That's fair. I can agree that it is a 1 dimensional subspace, and thus 1 dimensional. I guess I'm more focusing on the fact that it comes from a 2 dimensional space which is important information in saying that $mathbbR times 0$ isn't $mathbbR$
– Blake
Jul 23 at 17:10
I satisfied with the answer with Sir Blake, Sir I want to know R×0 is connected in R2.?
– Subhasish Chowdhury
Jul 24 at 7:33
 |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
2
down vote
So, you're asking a question which is a lot deeper than you might think. The short and sweet is that $mathbbR$ is one dimensional, but in $mathbbR^2$ you have 2 dimensions. So, $mathbbR times 0$ is a 1 dimensional subspace of a 2 dimensional space, the fact that it is a subspace of a 2 dimensional space is important information we'd like to keep. To get a better grasp of this concept, I suggest reading the book Flatland by Edwin Abbott Abbott.
Now, in a more technical sense, what you're asking about is something called a morphism. In our case, we would say, in plain english, that $mathbbR times 0$ is basically $mathbbR$. It behaves and acts just like it, however it is not $mathbbR$, it is a subset of $mathbbR^2$ that looks just like $mathbbR$. You would be incorrect, or at least remiss, to say that this is $mathbbR$.
Welcome to the beauty and elegance of Mathematics! :)
answered Jul 23 at 16:57
Blake
33219
$mathbbR times 0$ is 1 dimensional since a basis only has 1 element
– LinAlg
Jul 23 at 17:01
Well, it spans a 1 dimensional subspace of $mathbbR^2$, sure. However, it is still two dimensional because it lives within a two dimensional space. You can't tell me that it's 1 dimensional. How do you write an element of $mathbbR times 0$? Would you agree that it looks like $(x, 0) $?
– Blake
Jul 23 at 17:05
It is not two dimensional. The correct way of saying is it is a 1 dimensional subspace of a 2 dimensional space.
– LinAlg
Jul 23 at 17:06
That's fair. I can agree that it is a 1 dimensional subspace, and thus 1 dimensional. I guess I'm more focusing on the fact that it comes from a 2 dimensional space which is important information in saying that $mathbbR times 0$ isn't $mathbbR$
– Blake
Jul 23 at 17:10
I satisfied with the answer with Sir Blake, Sir I want to know R×0 is connected in R2.?
– Subhasish Chowdhury
Jul 24 at 7:33
 |Â
show 1 more comment
up vote
2
down vote
So, you're asking a question which is a lot deeper than you might think. The short and sweet is that $mathbbR$ is one dimensional, but in $mathbbR^2$ you have 2 dimensions. So, $mathbbR times 0$ is a 1 dimensional subspace of a 2 dimensional space, the fact that it is a subspace of a 2 dimensional space is important information we'd like to keep. To get a better grasp of this concept, I suggest reading the book Flatland by Edwin Abbott Abbott.
Now, in a more technical sense, what you're asking about is something called a morphism. In our case, we would say, in plain english, that $mathbbR times 0$ is basically $mathbbR$. It behaves and acts just like it, however it is not $mathbbR$, it is a subset of $mathbbR^2$ that looks just like $mathbbR$. You would be incorrect, or at least remiss, to say that this is $mathbbR$.
Welcome to the beauty and elegance of Mathematics! :)
answered Jul 23 at 16:57
Blake
33219
$mathbbR times 0$ is 1 dimensional since a basis only has 1 element
– LinAlg
Jul 23 at 17:01
Well, it spans a 1 dimensional subspace of $mathbbR^2$, sure. However, it is still two dimensional because it lives within a two dimensional space. You can't tell me that it's 1 dimensional. How do you write an element of $mathbbR times 0$? Would you agree that it looks like $(x, 0) $?
– Blake
Jul 23 at 17:05
It is not two dimensional. The correct way of saying is it is a 1 dimensional subspace of a 2 dimensional space.
– LinAlg
Jul 23 at 17:06
That's fair. I can agree that it is a 1 dimensional subspace, and thus 1 dimensional. I guess I'm more focusing on the fact that it comes from a 2 dimensional space which is important information in saying that $mathbbR times 0$ isn't $mathbbR$
– Blake
Jul 23 at 17:10
I satisfied with the answer with Sir Blake, Sir I want to know R×0 is connected in R2.?
– Subhasish Chowdhury
Jul 24 at 7:33
 |Â
show 1 more comment
up vote
2
down vote
up vote
2
down vote
So, you're asking a question which is a lot deeper than you might think. The short and sweet is that $mathbbR$ is one dimensional, but in $mathbbR^2$ you have 2 dimensions. So, $mathbbR times 0$ is a 1 dimensional subspace of a 2 dimensional space, the fact that it is a subspace of a 2 dimensional space is important information we'd like to keep. To get a better grasp of this concept, I suggest reading the book Flatland by Edwin Abbott Abbott.
Now, in a more technical sense, what you're asking about is something called a morphism. In our case, we would say, in plain english, that $mathbbR times 0$ is basically $mathbbR$. It behaves and acts just like it, however it is not $mathbbR$, it is a subset of $mathbbR^2$ that looks just like $mathbbR$. You would be incorrect, or at least remiss, to say that this is $mathbbR$.
Welcome to the beauty and elegance of Mathematics! :)
answered Jul 23 at 16:57
Blake
33219
So, you're asking a question which is a lot deeper than you might think. The short and sweet is that $mathbbR$ is one dimensional, but in $mathbbR^2$ you have 2 dimensions. So, $mathbbR times 0$ is a 1 dimensional subspace of a 2 dimensional space, the fact that it is a subspace of a 2 dimensional space is important information we'd like to keep. To get a better grasp of this concept, I suggest reading the book Flatland by Edwin Abbott Abbott.
Now, in a more technical sense, what you're asking about is something called a morphism. In our case, we would say, in plain english, that $mathbbR times 0$ is basically $mathbbR$. It behaves and acts just like it, however it is not $mathbbR$, it is a subset of $mathbbR^2$ that looks just like $mathbbR$. You would be incorrect, or at least remiss, to say that this is $mathbbR$.
Welcome to the beauty and elegance of Mathematics! :)
answered Jul 23 at 16:57
Blake
33219
edited Jul 23 at 17:10
answered Jul 23 at 16:57
Blake
33219
answered Jul 23 at 16:57
Blake
33219
answered Jul 23 at 16:57
Blake
33219
33219
$mathbbR times 0$ is 1 dimensional since a basis only has 1 element
– LinAlg
Jul 23 at 17:01
Well, it spans a 1 dimensional subspace of $mathbbR^2$, sure. However, it is still two dimensional because it lives within a two dimensional space. You can't tell me that it's 1 dimensional. How do you write an element of $mathbbR times 0$? Would you agree that it looks like $(x, 0) $?
– Blake
Jul 23 at 17:05
It is not two dimensional. The correct way of saying is it is a 1 dimensional subspace of a 2 dimensional space.
– LinAlg
Jul 23 at 17:06
That's fair. I can agree that it is a 1 dimensional subspace, and thus 1 dimensional. I guess I'm more focusing on the fact that it comes from a 2 dimensional space which is important information in saying that $mathbbR times 0$ isn't $mathbbR$
– Blake
Jul 23 at 17:10
I satisfied with the answer with Sir Blake, Sir I want to know R×0 is connected in R2.?
– Subhasish Chowdhury
Jul 24 at 7:33
 |Â
show 1 more comment
$mathbbR times 0$ is 1 dimensional since a basis only has 1 element
– LinAlg
Jul 23 at 17:01
Well, it spans a 1 dimensional subspace of $mathbbR^2$, sure. However, it is still two dimensional because it lives within a two dimensional space. You can't tell me that it's 1 dimensional. How do you write an element of $mathbbR times 0$? Would you agree that it looks like $(x, 0) $?
– Blake
Jul 23 at 17:05
It is not two dimensional. The correct way of saying is it is a 1 dimensional subspace of a 2 dimensional space.
– LinAlg
Jul 23 at 17:06
That's fair. I can agree that it is a 1 dimensional subspace, and thus 1 dimensional. I guess I'm more focusing on the fact that it comes from a 2 dimensional space which is important information in saying that $mathbbR times 0$ isn't $mathbbR$
– Blake
Jul 23 at 17:10
I satisfied with the answer with Sir Blake, Sir I want to know R×0 is connected in R2.?
– Subhasish Chowdhury
Jul 24 at 7:33
$mathbbR times 0$ is 1 dimensional since a basis only has 1 element
– LinAlg
Jul 23 at 17:01
$mathbbR times 0$ is 1 dimensional since a basis only has 1 element
– LinAlg
Jul 23 at 17:01
Well, it spans a 1 dimensional subspace of $mathbbR^2$, sure. However, it is still two dimensional because it lives within a two dimensional space. You can't tell me that it's 1 dimensional. How do you write an element of $mathbbR times 0$? Would you agree that it looks like $(x, 0) $?
– Blake
Jul 23 at 17:05
Well, it spans a 1 dimensional subspace of $mathbbR^2$, sure. However, it is still two dimensional because it lives within a two dimensional space. You can't tell me that it's 1 dimensional. How do you write an element of $mathbbR times 0$? Would you agree that it looks like $(x, 0) $?
– Blake
Jul 23 at 17:05
It is not two dimensional. The correct way of saying is it is a 1 dimensional subspace of a 2 dimensional space.
– LinAlg
Jul 23 at 17:06
It is not two dimensional. The correct way of saying is it is a 1 dimensional subspace of a 2 dimensional space.
– LinAlg
Jul 23 at 17:06
That's fair. I can agree that it is a 1 dimensional subspace, and thus 1 dimensional. I guess I'm more focusing on the fact that it comes from a 2 dimensional space which is important information in saying that $mathbbR times 0$ isn't $mathbbR$
– Blake
Jul 23 at 17:10
That's fair. I can agree that it is a 1 dimensional subspace, and thus 1 dimensional. I guess I'm more focusing on the fact that it comes from a 2 dimensional space which is important information in saying that $mathbbR times 0$ isn't $mathbbR$
– Blake
Jul 23 at 17:10
I satisfied with the answer with Sir Blake, Sir I want to know R×0 is connected in R2.?
– Subhasish Chowdhury
Jul 24 at 7:33
I satisfied with the answer with Sir Blake, Sir I want to know R×0 is connected in R2.?
– Subhasish Chowdhury
Jul 24 at 7:33
 |Â
show 1 more comment
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No, it is not a subspace at all.
– Sean Roberson
Jul 23 at 16:47
It depends on how you wish to think about $mathbbR$.
– Randall
Jul 23 at 16:48
$mathbb R$ is connected. It is homeomorphic to $mathbb R times 0 subset mathbb R^2$. You are done once you decide whether that is what you mean by "subspace".
– GEdgar
Jul 23 at 16:58