Baby Rudin examples 2.21: openness and closedness of subsets of $\mathbb{R}^2$ On page 33 of Principles of Mathematical Analysis by Walter Rudin(3rd Ed.), there are the examples:

2.21 Examples Let us consider the following subsets of $ \mathbb{R}^2 $:
(excluding the irrelevant parts)
(g) The segment $ (a,b) $.

I can't understand why Rudin says (g) can be regarded as a subset of $ \mathbb{R}^1 $. I know $ ((1,0),(2,0)) $ can be a subset of $ \mathbb{R}^1 $,  but what about $ ((1,2),(3,4)) $? Rudin also says the segment is not open if we regard it as a subset of $ \mathbb{R}^2 $, but is an open subset of $ \mathbb{R}^1 $. I understand such a segment is open in the real line, but how can it be close in $ \mathbb{R}^2 $?
This may be an old thread, but I am currently reading the book, so to speak I will give my $2$ cents:

If $(a,b)$ is a subset of $\mathbb{R}^1$, then the elements of $(a,b)$ are those $x$ for which we have
$$a &lt; x &lt; b , x\in \mathbb{R}^1$$

My idea for $\mathbb{R}^2$(probably not what Rudin meant):
If $(a,b)$ is a subset of $\mathbb{R}^2$, we can think of $(a,b)$ as an interval, just not being closed, that is:
then $a=(q,w), b=(z,r)$, that is $a$ and $b$ are from $\mathbb{R^2}$,
then if $x=(o,p)$ is an element of $(a,b)$, 
it follows that $q &lt; o &lt; z$ and also $w &lt; p &lt; r$.
That is we have a $2$-cell but we have removed it's borders.

From my understanding of the answers I've read so far, it seems Rudin thinks of something else.
Most likely the meaning is that we think of $(a,b)$ in $\mathbb{R}^2$ as of points of type $(x,0)$ for which $a &lt; x &lt; b$, that is $a,x,b$ are from $\mathbb{R}^1$, hence $(a,b)$ is a line in the plane and it follows that we can't have a neighbourhood in just a line, cause in $\mathbb{R}^2$ it's a circle instead.
You can think of the segment $ ((1,2),(3,4)) $ as a subset of the (infinite) line that passes through the points $ (1,2)$ and $ (3,4) $. Every such line is an $ \mathbb{R}^1 $.

You are right, it is not closed in $ \mathbb{R}^2 $. Remember that a set can be both not closed and not open.