Homeomorphic Image of Local Basis is Local Basis

Theorem
Let Let $T_\alpha = \left({S_\alpha, \tau_\alpha}\right)$ and $T_\beta = \left({S_\beta, \tau_\beta}\right)$ be topological spaces.

Let $\phi: T_\alpha \to T_\beta$ be a homeomorphism.

Let $s \in S_\alpha$.

Let $\BB$ be a local basis of $s$ in $T_\alpha$.

Then:
 * $\BB' = \set{ \phi \sqbrk B : B \in \BB}$ is a local basis of $\map \phi s$ in $T_\beta$