Let \(X\) be the infinite earring in \(\mathbb{R}^2\). (See Example 1 of §80.) Let \(C(X)\) be the subspace of \(\mathbb{R}^3\) that is the union of all line segments joining points of \(X \times 0\) to the point \(p = (0, 0, 1)\). It is called the cone on \(X\). Show that \(C(X)\) is simply connected, but is not locally simply connected at the origin.
Solution
The cone can be written as \[ C(X)=(X\times[0,1])/(X\times\{1\}), \] with cone point \(p\). The homotopy \[ H([x,t],s)=[x,(1-s)t+s] \] contracts all of \(C(X)\) to \(p\). Hence \(C(X)\) is contractible, in particular simply connected.
Let \(o=(0,0,0)\), the origin in the base \(X\times\{0\}\). We show \(C(X)\) is not locally simply connected at \(o\). Take the neighborhood \[ U=C(X)-\{p\}. \] Since the cone point is closed, \(U\) is open. Radial projection in the cone coordinate gives a deformation retraction \[ U\simeq X\times\{0\}=X. \]
Let \(V\subset U\) be any neighborhood of \(o\). Its intersection with the base contains a neighborhood of \(o\) in the infinite earring. Hence, for all sufficiently large \(n\), it contains the entire small circle \(C_n\) of the earring. Let \(\gamma_n\) traverse \(C_n\) once. Then \(\gamma_n\subset V\).
Under the deformation retraction \(U\to X\), the loop \(\gamma_n\) remains the standard loop on \(C_n\). Retraction of the earring \(X\to C_n\), collapsing all other circles to \(o\), sends it to a generator of \[ \pi_1(C_n,o)\cong\mathbb Z. \] Thus \(\gamma_n\) is not nullhomotopic in \(U\). Therefore no neighborhood \(V\subset U\) of \(o\) has all its loops nullhomotopic in \(U\); in particular there is no basis of simply connected neighborhoods at \(o\). Hence \(C(X)\) is not locally simply connected at the origin.