General Topology – The fundamental group of the "Introduction to Knot Theory" circle, Ralph H. Fox (1 & # 39;)


The book says at the beginning of discussing this title:

"Let the field of real numbers be denoted by R and the substitution of integers by $ J $. We denote the additive subgroup consisting of all integers that are multiples of 3 times $ 3J $. The circle, whose fundamental group we propose to calculate, can be considered as the group of factors $ R / 3J $ with the identification topology, that is, the largest topology such that canonical homomorphism $ phi: R rightarrow R / 3J $ It is a continuous mapping. A good way to imagine the situation is to consider $ R / 3J $ as a circle of circumference 3 mounted like a wheel on the real line $ R $ so you can roll freely from side to side without skidding. the possible points of tangency determine the $ many-one $ correspondence $ phi $"

And then the book began to try the following:

The image below $ phi $ of any open subset of $ R $ it is an open subset of $ R / $ 3 (5.2)

And the test is given below:

enter the description of the image here

My question:

I do not understand why $ phi ^ {- 1} phi (X) $ is given by this form, could someone please explain this to me?