Let \(X\) be the subspace of \(\mathbb{R}^2\) that is the union of the circles \(C_n\) of radius \(n\) and center \((n, 0)\) for \(n = 1, 2, \cdots\). Show that \(\pi_1(X)\) is the free group \(*_n \pi_1(C_n)\), the same as for the infinite wedge sum \(\bigvee_\infty S^1\). Show that \(X\) and \(\bigvee_\infty S^1\) are in fact homotopy equivalent, but not homeomorphic.
Solution
Let \(o=(0,0)\), the common tangency point of all the circles \(C_n\), and let \[ W=\bigvee_{n=1}^{\infty}S^1_n \] with the usual CW wedge topology.
1. There is a continuous map\[ f:W\to X \]whose restriction to \(S^1_n\) is a homeomorphism onto \(C_n\) taking the wedge point to \(o\).
Choose such a based homeomorphism separately on each circle. The CW wedge \(W\) has the weak topology with respect to its circles, so a map out of \(W\) is continuous when its restriction to every circle is continuous. Thus the circlewise maps assemble to a continuous \(f\).
2. There is a continuous map\[ g:X\to W \]such that each restriction\[ g|_{C_n}:C_n\to S^1_n \]has degree one.
Fix a small radius \(\varepsilon>0\). On each \(C_n\), collapse the connected arc \[ C_n\cap \overline{B_\varepsilon(o)} \] containing \(o\) to the wedge point, and map the complementary arc monotonically once around \(S^1_n\). This gives a continuous based degree-one map on each \(C_n\).
The definitions agree at the only common point \(o\). Moreover, \[ g\bigl(X\cap B_\varepsilon(o)\bigr) \] is exactly the wedge point. Hence \(g\) is continuous at \(o\), regardless of how small the prescribed neighborhoods of the wedge point are on the individual circles of \(W\).
If \(x\in X\setminus\{o\}\), then \(x\) lies on a unique circle \(C_n\) and has a sufficiently small Euclidean neighborhood in \(X\) meeting no other \(C_m\). Thus continuity at such \(x\) reduces to continuity of \(g|_{C_n}\).
3. The composite\[ g f:W\to W \]is homotopic to the identity on \(W\).
On each circle \(S^1_n\), the restriction \(gf|_{S^1_n}\) is a based map of degree one. Every based degree-one self-map of \(S^1\) is based-homotopic to the identity. Choose such a homotopy on each \(S^1_n\), fixing the wedge point throughout. Because \(W\) has the CW weak topology, these circlewise homotopies assemble to a continuous homotopy on \(W\).
4. The composite\[ fg:X\to X \]is homotopic to the identity on \(X\).
On each \(C_n\), the map \(fg\) is obtained by collapsing the short arc near \(o\) and reparametrizing the remaining arc once around \(C_n\). Homotope this reparametrization linearly in arc-length coordinates back to the identity, fixing \(o\).
Choose the parametrizations so that throughout the homotopy a point whose Euclidean distance from \(o\) is at most \(r<\varepsilon\) stays within distance at most \(2r\) of \(o\). This is possible by performing the reparametrization only along the two short end-arcs and using Euclidean arc length there. The resulting circlewise homotopies agree at \(o\), and the displayed uniform estimate gives continuity at \(o\) for the assembled homotopy. Away from \(o\), only one circle is locally present, so continuity is again circlewise. Hence \(fg\simeq\operatorname{id}_X\).
5. Therefore\[ X\simeq W. \]
The maps \(f\) and \(g\) are homotopy inverses by <1>3–<1>4.
6. Consequently\[ \boxed{ \pi_1(X,o) \cong *_{n=1}^{\infty}\pi_1(C_n,o) \cong *_{n=1}^{\infty}\mathbb Z. } \]
Homotopy equivalence gives \[ \pi_1(X,o)\cong\pi_1(W,*). \] The wedge-point neighborhoods in each circle deformation retract to the wedge point, so the wedge-sum form of van Kampen gives \[ \pi_1(W,*)\cong *_{n\ge1}\pi_1(S^1_n,*). \] Each factor is infinite cyclic.
7. The space \(X\) is first countable at \(o\).
It is a subspace of the metric space \(\mathbb R^2\). Thus the sets \[ X\cap B_{1/m}(o),\qquad m=1,2,\dots, \] form a countable neighborhood base at \(o\).
8. The infinite CW wedge \(W\) is not first countable at its wedge point.
Suppose \(U_1,U_2,\dots\) were a countable neighborhood base at the wedge point \(*\). For each \(n\), because \(U_n\cap S^1_n\) is a neighborhood of \(*\) in the \(n\)th circle, choose a point \[ p_n\in U_n\cap(S^1_n\setminus\{*\}). \] On each circle choose an open arc \(V_n\) about \(*\) that omits \(p_n\). The union \[ V=\bigcup_{n\ge1}V_n \] is a neighborhood of \(*\) in the CW wedge topology, since its intersection with every circle is open. But for every \(n\), \[ p_n\in U_n\setminus V, \] so \(U_n\nsubseteq V\). This contradicts the assumption that the \(U_n\) form a neighborhood base.
9. Hence \(X\) and \(W\) are not homeomorphic.
First countability is invariant under homeomorphism. By 7, \(X\) is first countable at the common point, while by 8 the wedge point of \(W\) is not.