Return to Answer

Despite the non-existence of a universal covering space, there is still an object that acts like a universal covering space for the Hawaiian earring. It is actually quite useful for understanding the structure of the fundamental group as a subgroup of the inverse limit $\varprojlim F_n$ of free groups. The "generalized universal covering" still provides all the same unique path lifting properties for maps out of locally path connected spaces that one might want. The study of these particular coverings for wild spaces is mainly due to Hanspeter Fischer and Andreas Zastrow. I think you should be able to use this to mimic the argument for graphs.

Let $\mathbb{H}$ be the Hawaiian earring and $\widetilde{\mathbb{H}}$ be the set of homotopy classes of paths starting at the basepoint. Take $p:\widetilde{\mathbb{H}}\to \mathbb{H}$, $p([\alpha])=\alpha(1)$ to be the endpoint projection and give $\widetilde{\mathbb{H}}$ the usual "whisker" topology that you do to construct universal covers: the basic neighborhoods are $B([\alpha],U)=\{[\alpha\cdot\epsilon]|\epsilon([0,1])\subset U\}$ where $U$ is open in $\mathbb{H}$. It turns out that $p:\widetilde{\mathbb{H}}\to \mathbb{H}$ is an open map which has unique lifting of all paths and homotopies of paths. Moreover, if $Y$ is path connected, locally path connected, and simply connected, then any based map $f:Y\to \mathbb{H}$ has a unique lift $\tilde{f}:Y\to \widetilde{\mathbb{H}}$ such that $p\tilde{f}=f$. Consequently, $p$ induces isomorphisms on higher homotopy groups.

It also turns out that $\widetilde{\mathbb{H}}$ has the structure of an $\mathbb{R}$-tree (uniquely arcwise connected metric space where each arc is isometric to a sub-arc of the reals), and these are known to be contractible (this is due to J. Morgan I believe).

Now you should have all the usual ingredients to mimic the usual argument for graphs. In fact, the Hawaiian earring is not special here. This should work out for all 1-dimensional Peano continua.

Despite the non-existence of a universal covering space, there is still an object that acts like a universal covering space for the earring space. It is actually quite useful for understanding the structure of the fundamental group as a subgroup of the inverse limit $\varprojlim F_n$ of free groups. The "generalized universal covering" still provides all the same unique path lifting properties for maps out of locally path connected spaces that one might want. The study of these particular coverings for wild spaces is mainly due to Hanspeter Fischer and Andreas Zastrow. I think you should be able to use this to mimic the argument for graphs.

Let $\mathbb{E}$ be the earring space, which is a shrinking wedge of a sequence of circles. Let $\widetilde{\mathbb{E}}$ be the set of homotopy classes of paths starting at the basepoint. Take $p:\widetilde{\mathbb{E}}\to \mathbb{E}$, $p([\alpha])=\alpha(1)$ to be the endpoint projection and give $\widetilde{\mathbb{E}}$ the usual "whisker" topology that you do to construct universal covers: the basic neighborhoods are $B([\alpha],U)=\{[\alpha\cdot\epsilon]|\epsilon([0,1])\subset U\}$ where $U$ is open in $\mathbb{E}$. It turns out that $p:\widetilde{\mathbb{E}}\to \mathbb{E}$ is an open map which has unique lifting of all paths and homotopies of paths. Moreover, if $Y$ is path connected, locally path connected, and simply connected, then any based map $f:Y\to \mathbb{E}$ has a unique lift $\tilde{f}:Y\to \widetilde{\mathbb{E}}$ such that $p\tilde{f}=f$. Consequently, $p$ induces isomorphisms on higher homotopy groups.

It also turns out that $\widetilde{\mathbb{E}}$ has the structure of an $\mathbb{R}$-tree (uniquely arcwise connected metric space where each arc is isometric to a sub-arc of the reals), and these are known to be contractible (this is due to J. Morgan I believe).

Now you should have all the usual ingredients to mimic the usual argument for graphs. In fact, the earring is not special here. This should work out for all 1-dimensional Peano continua.

Despite the non-existence of a universal covering space, there is still an object that acts like a universal covering space for the Hawaiian earring. It is actually quite useful for understanding the structure of the fundamental group as a subgroup of the inverse limit $\varprojlim F_n$ of free groups. The "generalized universal covering" still provides all the same unique path lifting properties for maps out of locally path connected spaces that one might want. The study of these particular coverings for wild spaces is mainly due to Hanspeter Fischer and Andreas Zastrow. I think you should be able to use this to mimic the argument for graphs.

Let $\mathbb{H}$ be the Hawaiian earring and $\widetilde{\mathbb{H}}$ be the set of homotopy classes of paths starting at the basepoint. Take $p:\widetilde{\mathbb{H}}\to \mathbb{H}$, $p([\alpha])=\alpha(1)$ to be the endpoint projection and give $\widetilde{\mathbb{H}}$ the usual "whisker" topology that you do to construct universal covers: Let $B([\alpha],U)=\{[\alpha\cdot\epsilon]|\epsilon([0,1])\subset U\}$ where $U$ is open in $\mathbb{H}$. It turns out that $p:\widetilde{\mathbb{H}}\to \mathbb{H}$ is an open map which has unique lifting of all paths and homotopies of paths. Moreover, if $Y$ is path connected, locally path connected, and simply connected, then any based map $f:Y\to \mathbb{H}$ has a unique lift $\tilde{f}:Y\to \widetilde{\mathbb{H}}$ such that $p\tilde{f}=f$. Consequently, $p$ induces isomorphisms on higher homotopy groups.

It also turns out that $\widetilde{\mathbb{H}}$ has the structure of an $\mathbb{R}$-tree (uniquely arcwise connected metric space where each arc is isometric to a sub-arc of the reals), and these are known to be contractible (this is due to J. Morgan I believe).

Now you should have all the usual ingredients to mimic the usual argument for graphs. In fact, the Hawaiian earring is not special here. This should work out for all 1-dimensional Peano continua.

Despite the non-existence of a universal covering space, there is still an object that acts like a universal covering space for the Hawaiian earring. It is actually quite useful for understanding the structure of the fundamental group as a subgroup of the inverse limit $\varprojlim F_n$ of free groups. The "generalized universal covering" still provides all the same unique path lifting properties for maps out of locally path connected spaces that one might want. The study of these particular coverings for wild spaces is mainly due to Hanspeter Fischer and Andreas Zastrow. I think you should be able to use this to mimic the argument for graphs.

Let $\mathbb{H}$ be the Hawaiian earring and $\widetilde{\mathbb{H}}$ be the set of homotopy classes of paths starting at the basepoint. Take $p:\widetilde{\mathbb{H}}\to \mathbb{H}$, $p([\alpha])=\alpha(1)$ to be the endpoint projection and give $\widetilde{\mathbb{H}}$ the usual "whisker" topology that you do to construct universal covers: the basic neighborhoods are $B([\alpha],U)=\{[\alpha\cdot\epsilon]|\epsilon([0,1])\subset U\}$ where $U$ is open in $\mathbb{H}$. It turns out that $p:\widetilde{\mathbb{H}}\to \mathbb{H}$ is an open map which has unique lifting of all paths and homotopies of paths. Moreover, if $Y$ is path connected, locally path connected, and simply connected, then any based map $f:Y\to \mathbb{H}$ has a unique lift $\tilde{f}:Y\to \widetilde{\mathbb{H}}$ such that $p\tilde{f}=f$. Consequently, $p$ induces isomorphisms on higher homotopy groups.

It also turns out that $\widetilde{\mathbb{H}}$ has the structure of an $\mathbb{R}$-tree (uniquely arcwise connected metric space where each arc is isometric to a sub-arc of the reals), and these are known to be contractible (this is due to J. Morgan I believe).

Now you should have all the usual ingredients to mimic the usual argument for graphs. In fact, the Hawaiian earring is not special here. This should work out for all 1-dimensional Peano continua.

Despite the non-existence of a universal covering space, there is still an object that acts like a universal covering space for the Hawaiian earring. It is actually quite useful for understanding the structure of fundamental group as a subgroup of the inverse limit $\varprojlim F_n$ of free groups. The "generalized universal covering" still provides all the same unique path lifting properties for maps out of locally path connected spaces that one might want. The study of these particular coverings for wild spaces is mainly due to Hanspeter Fischer and Andreas Zastrow. I think you should be able to use this to mimic the argument for graphs.

Let $\mathbb{H}$ be the Hawaiian earring and $\widetilde{\mathbb{H}}$ be the set of homotopy classes of paths starting at the basepoint. Take $p:\widetilde{\mathbb{H}}\to \mathbb{H}$, $p([\alpha])=\alpha(1)$ to be the endpoint projection and give $\widetilde{\mathbb{H}}$ the usual "whisker" topology that you do to construct universal covers: Let $B([\alpha],U)=\{[\alpha\cdot\epsilon]|\epsilon([0,1])\subset U\}$ where $U$ is open in $\mathbb{H}$. It turns out that $p:\widetilde{\mathbb{H}}\to \mathbb{H}$ is an open map which has unique lifting of all paths and homotopies of paths. Moreover, if $Y$ is path connected, locally path connected, and simply connected, then any based map $f:Y\to \mathbb{H}$ has a unique lift $\tilde{f}:Y\to \widetilde{\mathbb{H}}$ such that $p\tilde{f}=f$. Consequently, $p$ induces isomorphisms on higher homotopy groups.

It also turns out that $\widetilde{\mathbb{H}}$ has the structure of an $\mathbb{R}$-tree (uniquely arcwise connected metric space where each arc is isometric to a sub-arc of the reals), and these are known to be contractible (this is due to J. Morgan I believe).

Now you should have all the usual ingredients to mimic the usual argument for graphs. In fact, the Hawaiian earring is not special here. This should work out for all 1-dimensional Peano continua.

Despite the non-existence of a universal covering space, there is still an object that acts like a universal covering space for the Hawaiian earring. It is actually quite useful for understanding the structure of the fundamental group as a subgroup of the inverse limit $\varprojlim F_n$ of free groups. The "generalized universal covering" still provides all the same unique path lifting properties for maps out of locally path connected spaces that one might want. The study of these particular coverings for wild spaces is mainly due to Hanspeter Fischer and Andreas Zastrow. I think you should be able to use this to mimic the argument for graphs.

Let $\mathbb{H}$ be the Hawaiian earring and $\widetilde{\mathbb{H}}$ be the set of homotopy classes of paths starting at the basepoint. Take $p:\widetilde{\mathbb{H}}\to \mathbb{H}$, $p([\alpha])=\alpha(1)$ to be the endpoint projection and give $\widetilde{\mathbb{H}}$ the usual "whisker" topology that you do to construct universal covers: Let $B([\alpha],U)=\{[\alpha\cdot\epsilon]|\epsilon([0,1])\subset U\}$ where $U$ is open in $\mathbb{H}$. It turns out that $p:\widetilde{\mathbb{H}}\to \mathbb{H}$ is an open map which has unique lifting of all paths and homotopies of paths. Moreover, if $Y$ is path connected, locally path connected, and simply connected, then any based map $f:Y\to \mathbb{H}$ has a unique lift $\tilde{f}:Y\to \widetilde{\mathbb{H}}$ such that $p\tilde{f}=f$. Consequently, $p$ induces isomorphisms on higher homotopy groups.

It also turns out that $\widetilde{\mathbb{H}}$ has the structure of an $\mathbb{R}$-tree (uniquely arcwise connected metric space where each arc is isometric to a sub-arc of the reals), and these are known to be contractible (this is due to J. Morgan I believe).

Now you should have all the usual ingredients to mimic the usual argument for graphs. In fact, the Hawaiian earring is not special here. This should work out for all 1-dimensional Peano continua.