So I've just started learning a bit of topology and set theory and im getting confused on definitions of indexed family.
I've seen various websites and books describe an indexed family of sets like this
$\{A_{i}|i\in I \}$ is a function $S:I\longrightarrow A$ and we write $A_{i}$ instead of $S(i)$.
But on a post I saw on here I understood that an indexed family can contain repeated elements if this is so why do we use set builder notation to describe a family?
Shouldn't it be something more like $(U_i)_{i\in I}$ in this case the braces do not indicate a set.
Thanks in advance.
Best Answer
Unfortunately, both notations you describe are in use. Some people use $\{A_{i}\mid i\in I \}$ (or interchangeably, $\{A_i\}_{i\in I}$) to denote an indexed family, and others use $(A_i\mid i\in I)$ or $(A_i)_{i\in I}$ (there are also other variations you will occasionally see, like $\langle A_i\rangle_{i\in I}$). Personally, I strongly agree with you that the first notation doesn't make sense for an indexed family, since it should be describing a set, not a function. So I would say that $(A_i)_{i\in I}$ denotes the indexed family itself whereas $\{A_i\}_{i\in I}$ denotes the set of terms of the indexed family, i.e. the image of the function $I\to A$. However, this convention is not universal and some authors use $\{A_i\}_{i\in I}$ for the indexed family itself, or simply do not make a clear distinction between the indexed family and its image.