Let $Y=\mathbb{R}^2$ be the infinite square grid, i.e., the graph with vertices $\mathbb{Z}^2$ and edges representing all segments connecting points at distance $1$, which is the universal covering of $S^1 \vee S^1$. Let $A, B \subseteq X$ be the two copies of $S^1$, such that $X=A \cup B$ and $A \cap B={x_0}$. For the chosen covering map $p$, describe the induced coverings $p^{-1}(A) \to A$ and $p^{-1}(B) \to B$ (in terms of the classification of connected coverings of $S^1$).
The universal covering of $S^1 \vee S^1$ is: https://i.imgur.com/vfzIhRMl.png (sorry if it is written in italian)
Now if we consider for example the map $p^{-1}(A) \to A$ we have $p^{-1}(A)=\mathbb{R} \times\mathbb{Z}$ so why speaks it about connected coverings? Have i to consider only one of the copies of $\mathbb{R}$ of $\mathbb{R} \times\mathbb{Z}$?