The says in the beginning of discussing this title:
"Let the field of real numbers be denoted by R and the subring of integers by $J$. We denote the additive subgroup consisting of all integers which are a multiple of 3 by $3J$. The circle, whose fundamental group we propose to calculate, may be regarded as the factor group $R/3J$ with the identification topology, i.e., the largest topology such that the canonical homomorphism $\phi: R \rightarrow R/3J $ is a continuous mapping.A good way to picture the situation is to regard $R/3J$ as a circle of circumference 3 mounted like a wheel on the real line $R$ so that it may roll freely back and forth without skidding. the possible points of tangency determine the $many-one$ correspondence $\phi$"
And then the book started proving the following:
The image under $\phi$ of any open subset of $R$ is an open subset of $R/3$ (5.2)
My Questions:
1- I have seen the picture of the fundamental group of the circle from "Allen Hatcher" and I have seen the helix of the real line is covering the circle, so how is this related to the upper picture that this book has drawn?
2- What is the importance of proving this: ">The image under $\phi$ of any open subset of $R$ is an open subset of $R/3$"?
Could anyone help me to answer these questions please?
As you can see I am trying to compare the argument followed in "Allen Hatcher" with that followed in this book, so if anyone can give me a comparison, It will be greatly appeciated!