Is R is connected subspace of R2.?

Question

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.

Answer 1

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! :)