In Hatcher's book Page 79 I was asked to provide two-sheeted covering space $Y \rightarrow X_{1}$ such that the composition $Y \rightarrow X_{1} \rightarrow X$ of the two covering spaces is not a covering space. In here the space $X$ is the shrinking wedge of circles, and $X_{1}$ is placing infinite such spaces onto the line.

(See the figure in the book) http://www.math.cornell.edu/~hatcher/AT/ATpage.html

The example I imagined is this one:

I use $Y$ the same as $X_{1}$, but I make the mapping $Y\rightarrow X_{1}$ like this: I map and the second circle to the first circle, and map the rest to themselves. Then if $Y \rightarrow X_{1} \rightarrow X$ is a covering map, according to the defintion the inverse of a neighborhood in $X$, they must be disjoint; but in here they simply coincide.

I don't know whether this really works as he required. Mostly because the original space is sufficiently bad (not locally path connected) therefore I expect Hatcher would need me to utilize this condition. Also, I want to ask if one can assert that if $X$ is locally path connected, then $Y$ must be a covering space of X. I'm thinking about this because in the next page problem 16, Hatcher asked the reader to prove this:

"16. Give maps $X\rightarrow Y\rightarrow Z$ such that both $Y\rightarrow Z$ and the composition $X\rightarrow Z$ are covering spaces, show that $X\rightarrow Y$ is a covering space if $Z$ is locally path-connected...."

Sorry that this is a purely homework level question.