Covering map implies local homeomorphism

Statement
Suppose $$E$$ and $$B$$ are topological spaces and $$p:E \to B$$ is a fact about::covering map. Then, $$p$$ is a fact about::local homeomorphism.

Proof
Given: A covering map $$p:E \to B$$. A point $$e \in E$$.

To prove: There exists an open subset $$V$$ of $$E$$ containing $$e$$ such that the restriction of $$p$$ to $$V$$ is a homeomorphism.

Proof: