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:

My question:

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