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From Topology Without Tears:

Let $S$ be the collection of all the circles in the plane.If $S$ is a subbasis for a topology $\tau$ on $\mathbb{R}^2$.Describe the open sets in $(\mathbb{R}^2,\tau)$.

For two different circles,their intersection will be either one point or two points.So,considering intersection of all two possible circles that intersect at a single point,the basis will include singleton sets $\{r\} ,\forall r\in \mathbb{R}^2$

So,our topology is discrete topology.so,all subsets of $\mathbb{R}^2$ will be open.Am I correct?

Thanks in advance!

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Any point in the plane is the kissing point of two circles, i.e. their intersection. Looks good! Since the topology generated by a sub-basis is all unions of finite intersections, we are done.

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