Consider the space $\mathbb{R}^3$. Let $L$ and $P$ be the set of all lines and set of all planes in $\mathbb{R}^3$. Let $\tau _1$ and $\tau _2$ be topologies on $\mathbb{R}^3$ generated by subbases $L$ and $ P$ respectively.
Then what is the relation between $\tau _1$ and $\tau _2$? are they equivalent or one is finer than other.
My thought: Intersection of two lines give a point so $\tau _1$ is discrete topology. Intersection of planes gives us st. line. I thought $\tau _1$ is finer than $\tau _2$.
Intersection of three planes gives us a point, so $\tau_2$ is also discrete, and the two topologies are the same.
Note that usually, when we say one topology is "finer" than another, we don't exclude the possibility that they are equal. Thus you are right that $\tau_1$ is finer than $\tau_2$, even though they are both discrete.