Show that the subspace \(X \subset \mathbb{R}^3\) that is the union of the spheres \(S_n\) of radius \(1/n\) and center \((1/n, 0, 0)\) for \(n = 1, 2, 3, \dots\) is simply-connected.
Solution
Goal: Prove that the shrinking union of 2-spheres \(X = \bigcup_{n=1}^\infty S_n \subset \mathbb{R}^3\) meeting at the origin is simply connected (\(\pi_1(X, 0) = 0\)).
1. Geometry and topology of the space \(X\):Proof:
1.1. For each \(n \ge 1\), let \(S_n = \{x \in \mathbb{R}^3 : \|x - (1/n, 0, 0)\| = 1/n\}\) be the 2-sphere of diameter \(2/n\) passing through the origin \(0 = (0, 0, 0)\).
1.2. The spheres pairwise intersect only at the origin: \(S_n \cap S_m = \{0\}\) for all \(n \neq m\).
1.3. Since each \(S_n\) is path-connected and contains \(0\), the union \(X = \bigcup_{n=1}^\infty S_n\) is path-connected.
1.4. For each \(n \ge 1\), let \(p_n = (2/n, 0, 0) \in S_n\) denote the pole of \(S_n\) opposite to the origin.
2. Retraction of punctured spheres to the origin:Proof:
2.1. The punctured 2-sphere \(S_n \setminus \{p_n\}\) is homeomorphic to \(\mathbb{R}^2\) via stereographic projection from \(p_n\).
2.2. Hence there is a continuous deformation retraction \(H_n: (S_n \setminus \{p_n\}) \times [0, 1] \to S_n \setminus \{p_n\}\) that contracts \(S_n \setminus \{p_n\}\) to the origin \(0\), keeping \(0\) stationary:\[H_n(x, 0) = x, \quad H_n(x, 1) = 0, \quad H_n(0, t) = 0 \quad \text{for all } t \in [0, 1].\]
2.3. Extend \(H_n\) to the entire space \(X \setminus \{p_n\}\) by setting \(H_n(x, t) = x\) for all \(x \in S_m\) (\(m \neq n\)). Since \(S_n \cap S_m = \{0\}\) and \(H_n(0, t) = 0\), this extended map is continuous on \((X \setminus \{p_n\}) \times [0, 1]\).
2.4. Throughout the deformation \(H_n\), every point moves within \(S_n\), so \(\|H_n(x, t) - x\| \le \operatorname{diam}(S_n) = 2/n\) for all \(x \in X \setminus \{p_n\}\) and \(t \in [0, 1]\).
3. Perturbing loops to miss poles:Proof:
3.1. Let \(f: [0, 1] \to X\) be a continuous loop based at the origin: \(f(0) = f(1) = 0\).
3.2. For each \(n \ge 1\), the preimage \(f^{-1}(S_n \setminus \{0\})\) is an open subset of \((0, 1)\), hence a countable union of disjoint open intervals \(f^{-1}(S_n \setminus \{0\}) = \bigcup_j (a_{n,j}, b_{n,j})\).
3.3. On each interval \((a_{n,j}, b_{n,j})\), the path \(f\) is a path in \(S_n\) starting and ending at \(0\).
3.4. Since the 2-sphere \(S_n\) has dimension 2, any continuous path in \(S_n\) can be perturbed relative to its endpoints to miss the pole \(p_n\) (by the Simplicial Approximation Theorem or Stone–Weierstrass density of paths missing a point).
3.5. Applying this perturbation to each interval \((a_{n,j}, b_{n,j})\) produces a loop \(f_n\) homotopic to \(f\) whose image in \(S_n\) misses \(p_n\): \(p_n \notin f_n([0, 1])\).
4. Constructing the global null-homotopy:Proof:
4.1. Perturb \(f\) simultaneously on all spheres \(S_n\) so that \(f(I)\) misses every pole \(p_n\) for all \(n \ge 1\).
4.2. For each \(n \ge 1\), apply the deformation retraction \(H_n\) on \(S_n \setminus \{p_n\}\) from
2.
2.3. Concatenating these homotopies over all \(n \ge 1\) (rescaling the \(n\)-th homotopy to the time parameter interval \([1 - 2^{-(n-1)}, 1 - 2^{-n}]\)) defines a homotopy \(H: [0, 1] \times [0, 1] \to X\).
2.4. Continuity of \(H\) at \(t = 1\): For any \(\varepsilon > 0\), choose \(N > 2/\varepsilon\). For all stages \(n \ge N\), every point moves by at most \(\operatorname{diam}(S_n) = 2/n < \varepsilon\), and the image at stage \(N\) is contained in the \(\varepsilon\)-neighborhood \(\bigcup_{n \ge N} S_n \subset B(0, \varepsilon)\).
2.5. Thus \(H\) is a continuous homotopy in \(X\) between the loop \(f\) and the constant loop at \(0\).
2.6. Therefore \([f] = 0\) in \(\pi_1(X, 0)\), so \(\pi_1(X, 0) = 0\).
5. Conclusion:Proof:\(X\) is path-connected and \(\pi_1(X, 0) = 0\), so \(X\) is simply connected.