Need for basis for a topology Why did we need to define a basis for a topology?
I read that it is difficult to specify topology on a bigger set, so we define topology on a smaller collection.
Now I want to ask that what were the requirements to accomplish this plan?
When this question was in front of us, what actually we wanted to find?

The biggest advantage to talking about bases is that they control the entirety of a space's topological information while being easy to work with. The prototypical example would be something like open balls or open rectangles in $\mathbb{R}^n$. Open subsets of $\mathbb{R}^n$ can be very strange, but open balls are simple to imagine and easy to work with. If I want to, say, prove that some function $f: X \rightarrow \mathbb{R}^n$ is continuous I would, in principal, need to prove that $f^{-1}(U)$ is open for every single open subset of $\mathbb{R}^n$. However, in actuality, what I will typically prove is that $f^{-1}(B_r(x))$ is open for all $x$ and for all sufficiently small $r$. This is sufficient though, because preimages are well behaved with unions, and because I can represent any arbitrary open subset of $\mathbb{R}$ as a union of open balls.
