How to determine subsets of $\mathbb{R}^2$ open, closed, both, neither or compact?

Question

I need help with this problem:

What of the following subsets of $\mathbb{R}^2$ are (i) open, (ii) closed, (iii) both open and closed, (iv) neither open or closed, (v) compact?

1. $1<x_1<2$ and $-1<x_2<2$
2. $x_1^2+x_2^2>0$
3. $\Vert \mathbf{x}-(1,3) \Vert <1$
4. $x_1>x_2$

I now the definitions of closed, open and compact, but I'm cofused with both and neither. How do I show using the definitios if those subsets open, closed, etc? Hope you can help me.

Answer 1

Finite intesection of open sets is open. Also, if $f:X\rightarrow Y$ is continuous and $U$ is open, then $f^{-1}(U)$ is open.

For (1) consider $f_1(x_1,x_2) = x_1, f_2(x_1,x_2) = x_2$. Then $f_1^{-1}((1,2)) \cap f_2^{-1}((-1,2))$ is open. You can reason the same way for the other problems with continuous function ($g(x,y) = x^2+y^2, h(x,y)=\sqrt{(x-1)^2+(y-3)^2}, l(x,y) = x - y$).

In a connected space $X$ (like $\mathbb{R}^2$), the only open sets that are also closed are $X$ and $\emptyset$. In $\mathbb{R}^2$ the compacts are the closed and limited set, so when you determine that the sets are open, you can say (in this exercise) that those sets are not closed and not compact.

Answer 2

$(1)$ is the cartesian product of two open subsets of $\Bbb R$ and hence $(1)$ is an open subset of $\Bbb R^2,$ $(2)$ is the complement of the singleton $\{(0,0) \}$ in $\Bbb R^2.$ Since singletons in $\Bbb R^2$ are closed so $(2)$ is definitely open, $(3)$ is the open ball in $\Bbb R^2$ centered at $(1,3)$ and radius $1$ and since every open ball in $\Bbb R^2$ is an open set so $(3)$ is again an open set and finally $(4)$ is again open since it is the lower half of the straight line $y=x$ in $\Bbb R^2.$ Since neither of the regions given in the four options is closed so they cannot be compact as well by the converse of the Heine-Borel theorem in $\Bbb R^2.$

Answer 3

Of course neither of the sets defined above are both open and closed, but to find a set that possesses this property, we can think of $\Bbb R^n$ for any positive integer $n$ (which fulfills both definitions of closedness and openness). Also the set $C^0[0,1]$ (i.e. the set of functions continuous over $[0,1]$) is both closed and open with sup metric.