Let P be a partition such that P $=\{t_0,...,t_n\}$ over the interval $[a, b]$, then to refer to a point $i$ in the partition $P$, some would say $t_i$.
So my question would be then, in this case would $t_i$ be a function? In my understanding could you not say that the partition $P$ contains the ordered pairs $\{(0, t_0), (1, t_1),...,(n, t_n)\}$. Would this be enough to imply that $t_i$ is a function?
A partition, in this context, is taken to mean a finite subset $P$ of a real interval $[a,b]$ that contains the endpoints $a,b$ of the interval.
Given a real interval $[a,b]$ and a positive integer $n\in\mathbb{N}$, we can define a function $t\colon\{0,1,2,\ldots,n\}\to[a,b]$, and we can insist that the function is increasing so that $$a=t_{0}<t_{1}<\ldots<t_{n}=b.$$ The set you've written is not $P$, but is actually the graph of $t$. The partition $P$ itself is actually the image of $t$.
In fact, on closer inspection, we do this all the time: What I've just described is the rigorous justification for commonplace phrases like "let $A=\{a_{1},a_{2},a_{3},\ldots\}$".
So the answer is yes, you can think of $t$ as a function, and, equivalently, you can think of $P$ as the image of a function. It's just that people don't bother to make a big deal out of this.