I'm spending too much time thinking about this problem :
I need to show that the shrinking wedge of circles which is path connected, locally path connected ,doesn't have a simply connected covering space .
hatcher gives a condition for a space to be semi locally connected
it's about the induced injection between the fundamental group of a neighborhood U and the fundamental group of our space (here we need the opposite of that) .
I've asked my doctor about it and he told me that I can't use it since our space is "locally path connected " and not semi locally ..Can anyone help me prove that this X : shrinking wedge of circles does't have a simply connected covering space ?
I supposed it had , then i took a loop in my space X ,I lifted it up to a loop in the simply connected covering space , this loop would be homotopic to the constant loop, my intuition is to project it down , and prove that since my space is not simply connected this loop can't be trivial but I'm feeling there's something missing or wrong .. Any help is appreciated.
Theorem : A topological space $X$ has a simply connected covering space if and only if $X$ is path connected, locally path connected and semi-locally simply connected.
We are interested in the "only if" part of the story. Let $(E, e_0) \stackrel{p}{\to} (X, x_0)$ be a based covering map. Pick a neighborhood $U$ of $x_0$ that is evenly covered by $p$. $U_i$ be one of the slices of $p^{-1}(U)$, and let $f : S^1 \to X$ be a loop in $U_i$. Since, $p$ is a homeomorphism from $U_i$ to $U$, the loop $f$ lifts to a loop $\tilde{f}$ based at $e_0$. $E$ is simply connected, so there is a path-homotopy taking $\tilde{f}$ to the zero loop at $e_0$. Pushing down this path-homotopy to $X$ via $p$ gives the desired contraction of $f$ to the zero loop at $x_0$.
In the Hawaiian earring, it has the subspace topology of $\Bbb R^2$, hence any, however small, open neighborhood of the wedge point contains one of the circles in the boquet. Thus it's not semi-locally simply connected, which contradicts the existence of simply connected covering spaces.