I have this proposition,
Let X be a non-empty set and $\tau_1$, $\tau_2$ be topologies on X. Then $\tau_1 \subset \tau_2$ if and only if for every $ U \in \tau_1$ and $x \in U$ there exists $V = V_x \in \tau_2$ such that $x \in V$ and $V \subset U$.
I am not really getting a grasp for this proposition and I do not understand it intuitively, what is it trying to say?
We want every open set $U$ (under $\tau_1$) to be open in $\tau_2$ as well (in the inclusion), which means that every point of such $U$ should be an interior point with respect to $\tau_2$, and this is what the condition states exactly.