1
$\begingroup$

I don't understand this sentence;

The segment $(a,b)$ can be regarded as both a subset of $\mathbb{R}^2$ and an open subset of $\mathbb{R}^1$. If $(a,b)$ is a subset of $\mathbb{R}^2$, it is not open, but it is an open subset of $\mathbb{R}^1$.

What is 'segment $(a,b)$ in $\mathbb{R}^2$'?

  • 0
    No - I am pretty sure they mean the line from $a$ to $b$ along the $x$-axis.2012-07-23

2 Answers 2

3

That sentence is a good example of poor use of mathematical statements. I guess they meant the set $\{(x,y_0)\in\Bbb R^2\;:\;a< x< b\,\,,\,y_0\in\Bbb R\,\,\text{fixed}\}$

Taking $\,y_0=0\,$ above gives an interval on the x-axis in the plane...

1

An open set in the euclidean topology is an open interval. Equivalently we can look at "open balls": a set is open if, for all elements $x$ in the set, there is some neighborhood around $x$ that is still entirely contained in the set. For an interval $(a,b)$ in $\mathbb{R}$ this is true. In $\mathbb{R}^2$ this set of points would be the line from $(a,0)$ to $(b,0)$, non-inclusive. Here it is not true, because open balls in $\mathbb{R}^2$ have height as well as width. A line doesn't have any height whatsoever, so it doesn't contain any open balls and can't possibly be an open set itself.