Clarifications in definition of $\sigma$-algebra?

Question

I was reading the definition of $\sigma-$algebra and I wonder if $F=X$ ?

I think about this because in the definition we see that $F\in X$ and in $1.$ is the other contention.

But if I am correct, then 1. doesn't have so much sense, why do mathematicians mention this then?

Let $X$ be a set. Then a $\sigma-$algebra $F$ is a nonempty collection of subsets of $X$ such that

1. $X$ is in $F$

2. $A$ is in $F$, then $A^c$ in $F$

3. If $A_n$ is a sequence of elements of $F$, then $\bigcup A_n$ is in $F$.

Answer 1

You're confusing $\in$ and $\subset$ (or $\in$ and $\subseteq$). $F$ is a collection of subsets of $X$. That is, $F \subseteq P(X)$, the power set of $X$. "$X$ is in $F$" is an informal way of saying "$X\in F$".