What do we mean by $\bigcup_{i\in I}^{} T_i$ where $T_i$ are sets?

Question

What do we mean by $\bigcup_{i\in I}^{} T_i$ where $T_i$ are sets?

Actually my question is more about what do we mean by index set $I$? Is it like sequence? To see more clear it wouldn't hurt to take $I$ as $\mathbb N$ so $\bigcup_{i\in \mathbb N}^{} T_i$ would be $ T_1 \cup T_2 \cup T_3 \dots$

However I couldn't imagine other cases different from $\mathbb N$. Could you help me to figure out this?

Answer 1

You can't always take $I$ as $\mathbb{N}$, since $I$ could be uncountable, for example $I = \mathbb{R}$. If $I$ is countably infinite, you could take $I = \mathbb{N}$, but it might not be the most natural choice in the context. That's why there is a general $I$.

Answer 2

$I$ is the index set, which is defined as
$I:=\{i | T_i$ exists$ \}$
$I$ can be any set, it can be equal to $\mathbb N$ , $\mathbb R$ or $[0 ,1]$ etc.

Edit : I corrected the definition.

Answer 3

In analysis and topology you often want to associate sets with real numbers. For example, when considering some open neighborhood around $x$ - $U_x$. Then you often want to study some sort of union or intersection of the neighbourhoods, when $x \in A$ and $A$ may be uncountable.