Exercise 1.2.19 of Hatcher's Algebraic Topology

Question

I've been trying to prove exercise $1.2.19$ of Hatcher's algebraic topology:

Show that the subspace of $\mathbb{R}^3$ that is the union of the spheres of radius $\frac{1}{n}$ and center $(\frac{1}{n}, 0, 0)$ for $n = 1, 2, ···$ is simply-connected.

I was thinking of applying Seifert-Van Kampen inductively somehow, but I can't conclude using just that argument. I saw this proof but I would like to see a proof of this exercise without using $1-$skeletons.

How can I prove this using just Seifert-Van Kampen?

Answer 1

Here's one argument you could make. Let $p=(0,0,0)$, let $S_n$ be the sphere centered at $(1/n,0,0)$, and let $H_n=\bigcup_{k\geq n} S_k$. You want to show that $\pi_1(H_1,p)$ is trivial. Given a loop in $H_1$ based at $p$, use van Kampen to show that it can be homotoped into $H_2$ (split $H_1$ as the union of a small open neighborhood $U$ of $H_2$ and a contractible open set $V$ containing most of $S_1$, and then use the fact that $U$ deformation-retracts to $H_2$). Similarly, show that for each $n$, a loop in $H_n$ based at $p$ can be homotoped into $H_{n+1}$ without leaving $H_n$. Now given a loop in $H_1$ based at $p$, construct a homotopy as follows. From $t=0$ to $t=1/2$, homotope your loop into $H_2$. From $t=1/2$ to $t=2/3$, homotope your loop into $H_3$ without leaving $H_2$. From $t=2/3$ to $t=3/4$, homotope your loop into $H_4$ without leaving $H_3$. And so on. Finally, at $t=1$, take the constant loop at $p$. This homotopy is continuous at $t=1$ since for $t>(n-1)/n$, the image of the homotopy is contained in $H_n$.

Answer 2

Let $S_n$ be the sphere of radius $1/n$, and let $U_n$ be an open neighborhood of the origin $(0,0,0)$ within $S_n$. Then we can cover the space $X$ by $V_n=(\bigcup_{n=1}^\infty U_n)\cup S_n$ for $n=1,2,\cdots$. Since $V_n$ are path-connected and have trivial fundamental groups, the intersection of any two or more $V_n$'s is $\bigcup_{n=1}^\infty U_n$, which is also path-connected and has trivial fundamental group, $\pi_1(X)=*_n\pi_1(V_n)=0$.