Let $(X,O)$ be topological space which is defined by local basis(fundamental neighborhood system)$B_x$ at $x∈X$. Let $(X,O')$ be another topological space, which is defined by local basis $B'_x$ at $x∈X$.
If for all $x∈X, B_x'⊂B_x$ , then two topology coincidences ?
For example, open $ε$ ball and closed $ε$ ball defines the same topology.
Not in general; consider the case where $O$ is the discrete topology and $O'$ is any other topology. Then the condition will be true; any neighborhood of any point $x$ in the sense of $O'$ will also be a neighborhood of $x$ in the sense of $O$, because in this case any set containing $x$ is a neighborhood of $x$ in the sense of $O$.