An example of a set of points in $R^3$ which is not open, not closed, not convex and not bounded

Question

I can't seem to think of an example for this set. An example of a set of points in $R^3$ which is not open, not closed, not convex and not bounded

Answer 1

This works: $\mathbb{Q}^3$ ....

Answer 2

Take the open ball of radius 1, plus the line $(x,0,0)$, where $x \in \mathbb{R}$. Precisely, $X=\{(x,y,z): y=z=0 \text{ or } x^2+y^2+z^2<1 \}$. This is

• not bounded, because you have the infinite line;

• not open, because any ball around $(22,0,0)$ contain $(22,\epsilon,\epsilon)$ for some epsilon, which is not in $X$.

• not closed, because the succession $x_n = (0,0,1-(1/n))$ has limit $(0,0,1)$ which is not in X

• not convex, because the middle point between $(0,0,1/2)$ and $(2,0,0)$, i.e. $(1,0,1/4)$, it is not in X.

Answer 3

A symmetric example:

Let $B_r(p) = \{ x\in\mathbb R^3 : |x-p|\leq r \}$ and define $$X = \bigcup_{n\geq1} (B_{2n}(0)\setminus B_{2n-1}(0)).$$

Thus $X$ isn't open, nor closed, nor convex, nor bounded.