My professor wrote this:
"To study some topological property, we can always, without loss of generality, only focus on a basis for the topology."
Can someone explain this, and maybe give a simple example? I try an example below: If you want to prove some function $f\colon X \to Y$ is continuous, you don't have to take an arbitrary open set in $Y$ and show that its pre-image is open in $X$. Instead, you can take a basis element of $Y$'s topology and show its pre-image is open in $X$.