How to prove that it is a topology

Question

I have the set

$$\Omega_r=\{(x,y)\in\mathbb{R}^2, x^2+y^2=r^2\}, r\geq0$$

and i consider $\sigma$ the familly of $\emptyset$ and all union of the set $\Omega_r$

How to prove that it is a topology on $\mathbb{R}^2$ ?

It is clear that $\emptyset\in \sigma$ and $\mathbb{R}^2=\bigcup_{r\geq0}\Omega_r$

But if i take $O=\cup_{r\in L}\Omega_r$ and $S=\cup_{s\in S}\Omega_s$ what is $O\cap S$ please ?

Answer 1

It's simply $\bigcup _{r\in L\cap S}\Omega_r$ because the $\Omega_r$ are pairwise disjoint.

In fact, whenever you have a set $X$ and an equivalence relation on $X$, then there is a topology on $X$ where the open sets are precisely the unions of equivalence classes.

Answer 2

Your open sets are union of circles centered at the origin.

The intersections of such unions are what they have in common which is either empty set or another union of such circles.

Technically, you may write, $$ \bigg( \bigcup _{r\in L}\Omega_r \bigg)\bigcap \bigg(\bigcup _{r\in S}\Omega_r\bigg ) =\bigcup _{r\in L\cap S}\Omega_r$$