I am trying to understand the example 1.25 of Hatcher Book about the Srinking Wdge of Circle.
Consider the subspace $X$ of $\mathbb{R}^2$ that is the union of the circles $C_n$ of radius $\frac{1}{n}$ and center $\left(\frac{1}{n},0\right)$ for $ n = 1,2,3,\ldots $.
I have two question: why such retraction deffined by Hatcher exists? He define a family of retraction $r_n : X \to C_n$ collapsing all $C_i$'s except $C_n$ to the origin. I try to explicity this retraction, but I dont have sucess. Can anyone help me? Is possible given a explicy retraction for this?
If such retraction exist's, why this cannot be retract to a point, Since all $n-1$ circles are collapsed to the origin?
This retraction seems hard to believe. Since if we fix a $C_n$ and collapse all the other circles to the origin, I can't see what's stopping me from also collapsing the $C_n$ to the origin.