How does the natural topology work for $\mathbb{R}^2$? As we know, the natural (Euclidean) topology, $\tau$, on $\mathbb{R}^n$ is the topology generated by the base
$\mathcal{B}=\{B(p;r)\}$ where $B(p;r) = \{x \in \mathbb{R}^n \mid d(p,x) < r\}$. Taking unions of the open $B(p;r) \in \mathcal{B}$, we can construct any set in $\mathbb{R}^n$.
On the real number line, $\mathbb{R},$ this is obvious. Let $(a,b)$ and $(c,d)$ be intervals on $\mathbb{R}$. Then if $(a,b)$ and $(c,d)$ are disjoint, then $(a,b)\cap(c,d)=\varnothing \in \tau$ and $(a,b)\cup(c,d) \in \tau.$ If $(a,b)$ and $(c,d)$ are not disjoint, then $(a,b)\cap(c,d)=(c,b)$ [assuming that $a<c$ and $b<d$]. But this $(c,b) \in \tau$ so it still works. With $\mathbb{R}$, it's easy to understand how the unions and intersections of 1-dimensional intervals on $\mathbb{R}$ satisfies our topology.
But when we get to 2 dimensions, things get a little harder. Suppose we have $B(p_1;r)$ and $B(p_2;r) \in \mathbb{R}^2$. What open ball in $\mathbb{R}^2$ would represent their intersection? I've attached a small illustration below to help give an idea of what I mean.

So we have a blue disc and a red disc. Their intersection is represented by the overlapping, purple area.
How do we represent the purple region as an open ball?

Solution Idea:
The purple region needn't be an open ball. It can be the union of many open balls. What if we represented $B(p_1;r) \cap B(p_2;r)$ as the union of smaller open balls such that the union of the open balls "fills" the purple region?
 A: The natural topology on $\mathbb{R}^n$ is NOT the one you describe. What you describe does not satisfy the axioms of a topology. This is not even true for $n=1$. How would you describe $]0,1[\cup ]2,3[$ as an interval ? Answer: you can't since an interval is connected while the union of two disjoint intervals is not.
What you described is a BASIS for the topology (see https://en.wikipedia.org/wiki/Base_(topology) ).
If you don't like the notion of basis, in the particular case of $\mathbb{R}^n$ (or a metric space), you can describe the open sets as the sets $U$ such that for all $x\in U$, $U$ contains a ball centered in $x$.
So for you purple intersection, it is now more or less clear on the picture that if $x$ lies in the purple region, you can draw a small ball around $x$ contained in this purple region. It is a good exercise to formalize it with the definitions of open balls.
A: The open sets in the standard topology on $\mathbb{R}^2$ don't necessarily have to be represented by open balls. For a subset $U\subseteq\mathbb{R}^2$ to be open in the standard topology, it's only necessary that for all $x\in U$ there exists an open ball $B$ such that $x\in B$ and $B\subseteq U$. I'll leave proving this is the case for finite intersections and arbitrary unions as an exercise for you.
