Can someone show me an example of an space $X$ non locally connected and another space $X'$ such that $\varpi: X' \to X$ is not a covering but $\varpi: \varpi^{-1}(X_i) \to X_i$ is, for each connected component $X_i \subset X$?
I've been trying to use as $X$ something as $\{0\} \cup \{\frac{1}{n} \}_{n \in \mathbb{N}}$ but to no avail. Is there any canonical space to use as a counterexample for questions like this?
consider $A= \{0\}\cup \{1/n|n\in \mathbb{N}\}$... define a map $p:A \to \mathbb{N}\cup \{0\}$ , $p(0)=0$ and $ p(1/n)=n $ ...then it is not a covering map, since $p$ is not even continuous , but restricting to each component gives rise to a bijection, which is homeomorphsim , in other word covering map.