Base and neighborhoods Mathematical Analysis II by Zorich:

Definition 1 A set $X$ is said to be a topological space if a system $\tau$ of subsets of $X$ is exhibited (called open sets in $X$) possessing the following properties:
a) $\emptyset\in\tau$;$X\in\tau$.
b) $(\forall\alpha\in A(\tau_\alpha\in\tau))\implies\bigcup_{\alpha\in A}\tau_\alpha\in\tau$.
c) $(\tau_i\in\tau;i=1,\dots,n)\implies\bigcap^n_{i=1}\tau_i\in\tau.$


Definition 3 A base of the topological space $(X,\tau)$ is a family $\mathfrak{B}$ of open subsets of $X$ such that every open set $G\in\tau$ is the union of some collection of elements of the family $\mathfrak{B}$.


Definition 5 A neighborhood of a point of a topological space $(X,\tau)$ is an open set containing the point.

I want to ask:

It is clear that the system of all neighborhoods of all possible points of topological space can serve as a base for the topology of that space.

If $G$ is an nonempty open set in $X$, then it contains some point $p\in X$ and is a neighborhood of $p$. Hence the collection of all neighborhoods of all possible points of a topological space $(X,\tau)$ is $\tau\setminus\emptyset$. It this what the author means? I think this statement expresses nothing.

You are right, the system $\mathfrak N$ of all neighborhoods of all points of a topological space $(X,\tau)$ is a base for the topology $\tau$. In fact, $\mathfrak N = \tau \setminus \{\emptyset\}$.
Let me remark $\emptyset$ can always be written as the union over the empty index set. For each index function $\iota : A \to \mathcal B \subset \tau$ into the base $\mathcal B$, we define $\tau_\alpha = \iota(\alpha)$ and $\bigcup_{\alpha \in A} \tau_\alpha = \{ x \in X \mid \exists \alpha \in A : x \in \tau_\alpha\}$. If $A = \emptyset$, we have only one (trivial) function $\emptyset \to \mathcal B$, and the above general definition gives us $\bigcup_{\alpha \in \emptyset} \tau_\alpha = \emptyset$.
Anyway, the system  $\mathfrak N$ is only one of many bases. The intention of the concept of "base" is to find smaller systems than $\tau$ or $\mathfrak N$ which generate $\tau$ by forming unions. In other words, we want to make an "economical" approach.
For example, in $\mathbb R^n$ with its standard Euclidean topology the open balls
$$B(x,\epsilon)  = \{ x' \in \mathbb R^n \mid \lVert x' - x \rVert < \epsilon \}$$
form a base.
