Homeomorphic Image of Neighborhood Basis is Neighborhood Basis

Theorem
Let Let $T_\alpha = \struct{S_\alpha, \tau_\alpha}$ and $T_\beta = \struct{S_\beta, \tau_\beta}$ be topological spaces.

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

Let $s \in S_\alpha$.

Let $\NN$ be a neighborhood basis of $s$ in $T_\alpha$.

Then:
 * $\NN' = \set{ \phi \sqbrk N : N \in \NN}$ is a neighborhood basis of $\map \phi s$ in $T_\beta$