The following definitions are taken from Zorich's book and I am fairly familiar with them, but I noticed one interesting point that I had not noticed before.
Definition 1. A set $\mathcal{B}$ of subsets $B\subset X$ of a set $X$ is called a base in $X$ if the following conditions hold:
$\forall B\in \mathcal{B} \ (B\neq \varnothing)$;
$\forall B_1\in \mathcal{B} \ \forall B_2\in \mathcal{B} \ \exists B\in \mathcal{B} \ (B\subset B_1\cap B_2)$
Definition 2. We shall say a certain property of functions or a certain relation between functions holds ultimately over a given base $\mathcal{B}$ if there exists $B\in \mathcal{B}$ on which it holds.
Definition 3. Let us agree that the notation $f=O(g)$ over the base $\mathcal{B}$ means that the relation $f(x)=\beta(x)g(x)$ holds ultimately over $\mathcal{B}$ where $\beta(x)$ is ultimately bounded over $\mathcal{B}$.
Let us write Definition 3 in more detail. The notation $f=O(g)$ over the base $\mathcal{B}$ means that $\exists \beta:X\to \mathbb{R}$ and $\exists B_1\in \mathcal{B}$ such that $\forall x\in B_1$ we have $f(x)=\beta(x)g(x)$ and $\exists B_2\in \mathcal{B}$ such that $\forall x\in B_2$ we have $|\beta(x)|\leq C$.
By definition of base it follows that $\exists B_3\subset B_1\cap B_2$ and we notice that $\forall x\in B_3$ we have $|f(x)|=|\beta(x)||g(x)|\leq C|g(x)|$.
Can we summarize it as follows?
If $f(x)=O(g(x))$ over the base $\mathcal{B}$, then $\exists C>0 \ \exists B_3\in \mathcal{B}: \forall x\in B_3 \ (|f(x)|\leq C|g(x)|)$
I don't think that we can do it because the constant $C>0$ depends on $B_3$ and $B_3$ depends on $\beta$.
For example, Wikipedia says that $f(x)=O(g(x))$ as $x\to +\infty$ if there exists a positive real number $M$ and a real number $x_0$ such that $$|f(x)|\leq M|g(x)| \quad \text{for all} \quad x\geq x_0.$$
I'd be very grateful if you can explain my question please! Thank you!