I am reading the stochastic differential equations and came across some definitions.
Path of X : for $\omega \in \Omega$ $X(.,\omega) : I \to S $
What I understood is that it is that, we fix one $\omega$ and calculate the value for each $X_t$ so the elements will be of the form $(1,X_1(\omega)) $ , $(2,X_2(\omega))$ ...... so if we consider I as time axis then it will be a plot between time and $X(\omega)$. And as we change $\omega $ we get different plots. Is this correct?
Path space : $ F \subset S^I$ such that for all $\omega \in \Omega $, $X(.,\omega) \in F$
Here my previous understanding is a bit contradictory since the elements in $S^I$ will be of the form $(S_1, S_2,....)$ an index(I) dimensional space. Where am I wrong in these definitions? If possible can you explain with some figures?
And finally does $\sigma^{I} $ same as the product sigma algebra generated by $\sigma_1 \times \sigma_2 \times ....$
Path of X
For each $\omega$ the path of $X$ is a function from $I$ to $S$. In particular it is the function $f(t)=X(t,\omega)$ (with $\omega$ fixed). You seem however to be confusing the path (which is a function) with the graph of the path (whose elements are on the form $(t,f(t)))$. Functions and graphs are closely related, but not the same thing.
Path space
Usually the notation $S^I$ refers to the set of all functions $f$ from the set $I$ to the set $S$, and in particular for each $\omega \in \Omega$ we have that the path $t\mapsto X(t,\omega)$ is an element of the path space $S^I$. Sometimes we want to restrict the path space to only allow certain types of functions, for instance we may restrict the path space to only allow continuous functions, and thus instead of working with the full path space $S^I$ we could work with a smaller path space, such as $F=C^0(I)\subseteq S^I$.
Countable vs. Uncountable case
It seems to me, that you have mostly concerned yourself with the countable case, since expressions such as $(S_1,S_2, \dots)$ or $\sigma_1 \times \sigma_2 \times \dots$ have no meaning if $I$ is uncountable. However when $I$ is countable, then there is a natural identification between the set $S^I$ (as i defined it above) and the set of all sequences indexed by $I$. But as stochastic calculus and stochastic differential equations are concerned, it is mostly the uncountable case that matters, and thus i would encourage that you get used to thinking of paths as functions rather than sequences.
$\sigma$-algebra
Recall that the product $\sigma$-algebra on $\mathbb{R}^2$ is the smallest $\sigma$-algebra, which makes the coordinate projections $(x,y)\mapsto x$ and $(x,y)\mapsto y$ measurable. The concept is similar for $\sigma^I$, which can be defined as the smallest $\sigma$-algebra, which makes the coordinate projections \begin{align*} \pi_t: S^I &\rightarrow S \\ f &\mapsto f(t) \end{align*} measurable for all $t \in I$. When $I$ is countable, this corresponds to the sigma algebra generated by $\sigma \times \sigma \times \dots$, but as mentioned before, this does not even make sense when $I$ is uncountable.
Why do we care about path spaces?
We have just defined the path-space $S^I$ and the corresponding $\sigma$-algebra $\sigma^I$. This means, that $(S^I,\sigma^I)$ is a measurable space and it is thus possible to speak of $S^I$-valued random variables, i.e. measurable functions $\mathbf{Y}:\Omega \rightarrow S^I$. Try to verify for yourself, that $\mathbf{Y}:\Omega \rightarrow S^I$ is $\sigma^I$ measurable if and only if $\pi_t(\mathbf{Y})$ is measurable for all $t\in I$ in particular we have the following:
$$(X_t)_{t\in I} \text{ is an $S$-valued stochastic proces } \Leftrightarrow \mathbf{X} \text{ is } \sigma^I \text{-measurable},$$ where $\mathbf{X}:\Omega \rightarrow S^I$ is defined as $$\mathbf{X}(\omega) = X(\cdot , \omega) \quad \text{(This is the path of $X$ as defined above)}.$$ Because of this last characterisation, some authors actually prefer to think of stochastic processes as $S^I$-valued random variables, rather than collections of $S$-valued random variables.