To find closure and interior of a subset of plane.

Question

Find the interior and closure of $C\subset \mathbb R^2$ where $C=A\cup B$

A=$\{(x,y)\in\mathbb R^2:\ y=0\}$

B=$\{(x,y)\in\mathbb R^2:\ x\gt 0 , y\ne 0\}$.

I found their union to be the I and IV quadrant of plane. So $\text{int}(C) =\{(x,y)\in\mathbb R^2:\ x\gt 0\}$. What about closure?

Answer 1

The closure is similar to the snowshovel. Draw a picture.

Answer 2

The closure is $C\cup D$, where $D=\{(x,y)|x=0\}$ is the set of limit points of $C$ that are not in $C$.