# Fundamental group of shrinking wedge of spheres.

Let $X$ be the subspace of $\mathbb R^3$ which is union of the spheres of radius $1/n$ and centered at $(1/n,0,0)$.Then $X$ is simply connected.

I had thought for it in this way to attach 3-cells to single point namely origin but then I realize the space I will get is wedge sum of spheres not the space given in the question.

Please help to figure out the fundamental group of the space in question

Not a homework problem

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It isn't important at all if your question is homework or not: the tag only serves for purposes of how deep a hint is given or not. People will still try to help you. – DonAntonio Dec 30 '12 at 13:28
But $\pi_{1} (S^{2})=0$,so how are you getting $\mathbb Z$? – Shraddha Srivastava Dec 30 '12 at 13:35
The only thing you need to worry about is a loop that is on infinitely many of the spheres. But it seems to me you ought to be able to define a homotopy by being clever enough. – JSchlather Dec 30 '12 at 13:37
@ShraddhaSrivastava, I thought we were working with copies of $\,S^1\,$ instead... – DonAntonio Dec 30 '12 at 13:39
After some homotopy, we can assume that the loop you are looking at avoid at least one point on each sphere. Deleting one point from each sphere makes the space contractible. – Andrew Dec 30 '12 at 13:42
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The only problem is to prove that any loop $f:I\rightarrow X$ that passes through the origin for infinitely many times is nullhomotopic. Here is the idea:
Let $X=\bigvee_n S_{1/n}^2$ and $O$ be the origin, then the subset $E:=f^{-1}(X\backslash\{O\})$ should be open in $I=[0,1]$. Therefore $E$ can be written as disjoint union of infinitely many open intervals , i.e. $E=\bigcup_{i=1}^{\infty}(a_i,b_i)$. Then we can define a sequence of homotopies $F_i:I\times [1-\frac{1}{i},1-\frac{1}{i+1}]\rightarrow X$, such that $F_i(\cdot,1-\frac{1}{i})=f_i$ and $F_i(\cdot,1-\frac{1}{i+1})=f_{i+1}$, where $f_i$ is the loop we get after having shrunk the first $i-1$ small loops $f((a_k,b_k))$ to $O$. Note that $f_1=f$. Finally we can construct a homotopy $F:I\times I\rightarrow X$ by gluing $F_i$ together, i.e. $F(\cdot,t)=F_i(\cdot,t)$ if $t\in[1-\frac{1}{i},1-\frac{1}{i+1}]$ and $F(\cdot,t)=O$ if $t=1$. By pasting lemma, $F$ is continuous whenever $t<1$. To see that $F$ is also continuous when $t=1$, it suffices to prove that for any neighborhood $U$ of the origin $O$ in $X$, $f^{-1}(U)$ contains all but finitely many intervals of $(a_i,b_i)$. This is true because all the intervals $(a_i,b_i)$ form a disjoint open cover of $f^{-1}(X\backslash U)$, which is compact in $I$.