It is an extension of another question:
Distinct topologies making a map a local homeomorphism
If $X$ is a topological space and $p:E\rightarrow X$ is a function, is it possible for $E$ to have non-homeomorphic topologies each one making $p:E\rightarrow X$ a local homeomorphism?
Obs.: A local homeomorphism $f:A\rightarrow B$ between topological spaces is a function such that for every $a\in A$ there are open sets $U$ and $V$ in $A$ and $B$ such that $a\in U$ and $f\upharpoonright U:U\rightarrow V$ is a homeomorphism.