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.
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No, it is not a subspace at all. – Sean Roberson Jul 23 '18 at 16:47
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It depends on how you wish to think about $\mathbb{R}$. – Randall Jul 23 '18 at 16:48
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$\mathbb R$ 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 '18 at 16:58
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So, you're asking a question which is a lot deeper than you might think. The short and sweet is that $\mathbb{R}$ is one dimensional, but in $\mathbb{R}^2$ you have 2 dimensions. So, $\mathbb{R} \times \{0\}$ 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 $\mathbb{R} \times \{0\}$ is basically $\mathbb{R}$. It behaves and acts just like it, however it is not $\mathbb{R}$, it is a subset of $\mathbb{R}^2$ that looks just like $\mathbb{R}$. You would be incorrect, or at least remiss, to say that this is $\mathbb{R}$.
Welcome to the beauty and elegance of Mathematics! :)
Blake
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$\mathbb{R} \times {0}$ is 1 dimensional since a basis only has 1 element – LinAlg Jul 23 '18 at 17:01
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Well, it spans a 1 dimensional subspace of $\mathbb{R}^2$, 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 $\mathbb{R} \times {0}$? Would you agree that it looks like ${(x, 0) | x \in \mathbb{R}}$? – Blake Jul 23 '18 at 17:05
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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 '18 at 17:06
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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 $\mathbb{R} \times {0}$ isn't $\mathbb{R}$ – Blake Jul 23 '18 at 17:10
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I satisfied with the answer with Sir Blake, Sir I want to know R×{0} is connected in R2.? – Subhasish Chowdhury Jul 24 '18 at 07:33
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@SubhasishChowdhury well, it depends on what your notion of connected is. If you mean topologically, then you need to ask if it is the union of two disjoint, non empty sets. If it isn’t, then we consider it to be connected :) – Blake Jul 24 '18 at 14:41