The book continues defining a set $C_{n}$ for every integer $n$ to be the image under $\phi$ of the open interval $( n-1, n+1)$. It said that:"It follows from $(5.2)$ that each $C_{n}$ is open and from the above remarks that the mapping $$\phi_{n} : (n-1, n+1) \rightarrow C_{n}$$
defined by setting $\phi _{n} (x) = \phi (x), n-1 < x < n+1,$ is a homeomorphism. The sets $C_{n}$ form an open cover of the circle. However, this cover consists only of 3 distinct sets because, as easily shown, $$C_{n} = C_{m} \ iff \ \phi (n) = \phi (m)$$ (Which is not easily shown for me, could anyone explain this for me please?) "
Then the book added:
My questions:
1-I do not understand the function of $\phi_{n}$, are they covering maps? 2-Also, I do not understand why properties $(ii)$ & $(iii)$ are important properties? why are those properties required?
Could anyone help me in answering this questions please?
EDIT:
"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)
Your book has a somewhat unconventional approach to determine the fundamental group of the circle. Usually this is done within the theory of covering projections (by considering the covering projection $e : \mathbb R \to S^1, e(t) = e^{2\pi i t}$). You book avoids to develop a general thery of covering projections, but only considers the quotient map $\phi : \mathbb R \to S = \mathbb R / 3\mathbb Z \approx S^1$a and gives handwoven proofs to compute $\pi_1(S^1)$. There is a great overlap with the general theory of covering projections, but perhaps this is not clearly visible. My recommendation is therefore to consult another book dealing with covering projections and treating the fundamental group of the circle as a simple application.
Here are answers to your questions.
For $k \in \mathbb Z$ let $T_k : \mathbb R \to \mathbb R, T_k(x) = x + 3k$, be the translation by $3k$. Each $T_k$ is a homeomorphism (with inverse $T_{-k}$).
We have $\phi(x) =\phi(y)$ iff $x - y \in 3\mathbb Z$, i.e. $x = y + 3k$ for some $k \in \mathbb Z$. In other words, $\phi(x) =\phi(y)$ iff $x = T_k(y)$ for some $k$. In particular, $\phi \circ T_k = \phi$ for all $k$.
Thus for any $M \subset \mathbb R$ $$\phi^{-1}(\phi(M)) = \bigcup_{k \in \mathbb Z} T_k(M) .$$ To see this observe that $x \in \phi^{-1}(\phi(M)) \Leftrightarrow \phi(x) \in \phi(M) \Leftrightarrow \phi(x) = \phi(y)$ for some $y \in M \Leftrightarrow x = T_k(y)$ for some $y \in M$ and some $k \in \mathbb Z \Leftrightarrow x \in \bigcup_{k \in \mathbb Z} T_k(M)$.
This shows that $\phi$ is an open map (because for open $U \subset \mathbb R$ all $T_k(U)$ are open).
Moreover each open interval $(a,b)$ of length $b -a \le 3$ is mapped bijectively onto $\phi((a,b))$. Hence $\phi_{(a,b)} : (a,b) \stackrel{\phi}{\rightarrow} \phi((a,b))$ is a homeomorphism. This shows that the $\phi_n : (n-1,n+1) \to C_n$ are homeomorphisms onto open $C_n \subset S$.
It is easy to see that $C_0, C_1, C_2$ are three distinct sets. In fact $C_0$ contains $p_0$, but not $p_1, p_2$, $C_1$ contains $p_1$, but not $p_0, p_2$, and $C_2$ contains $p_2$, but not $p_0, p_1$. Moreover, any two of them have nonempty intersection and $C_0 \cup C_1 \cup C_2 = S$.
If $\phi(n) =\phi(m)$, then $n = T_k(m)$ for some $n$. Hence $C_m = \phi((m-1,m+1)) = \phi(T_k(m-1,m+1))) = \phi((T_k(m)-1,T_k(m)+1)) = C_{T_k(m)} = C_n$. Conversely, if $C_m = C_n$, then we find $m_0, n_0 \in \{0,1,2\}$ such that $\phi(m) = \phi(m_0)$ and$\phi(n) = \phi(n_0)$. Hence $C_{m_0} = C_{n_0}$ and we conclude $m_0 = n_0$. This shows $\phi(m) = \phi(n)$.
The $\phi_n$ are homeomorphism, in particular covering maps. But that is irrelevant here. We know that $$\phi^{-1}(C_n) = \bigcup_{k\in \mathbb Z}T_k((n-1,n+1))$$ where the $T_k((n-1,n+1))$ are pairwise disjoint open intervals which are mapped by $\phi_{T_k(n)}$ homeomorphically onto $C_n$. This shows that $\phi$ is a covering map.
To understand the purpose of properties (ii) and (iii) you have to finish reading the proofs in your book. You will see that these properties are used at several places. But recall my above recommendation.