I am reading "Topology without tears " book, and got confused about proposition 2.3.4 page 60:
$\tau_1 =\tau_2$ iif
1) for each $B \in \mathscr B_1 $ and $ \forall x \in B , \exists B' \in \mathscr B_2 $ such that $ x\in B' \subseteq B $
2) for each $B \in \mathscr B_2 $ and $ \forall x \in B , \exists B' \in \mathscr B_1 $ such that $ x\in B' \subseteq B $
I need intuition about this proposition! (If only part)
We are trying to prove two topologies are equivalent, so first we prove $\tau_1$ is subset of $\tau_2$ . Stop! Please explain what we are doing with basis of topologies here? Then book says by definition of basis and consideration of (1) above B is open in $\tau_2 $. Why?
Then book concludes $\mathscr B_1$ is $\subseteq \tau_2 $. Why?
Please expand your explanation part by part.