| (i) | $\mathcal{P}(X)$ is the D80: Power set of $X$ |
| (1) | \begin{equation} \emptyset \in \mathcal{F} \end{equation} |
| (2) | \begin{equation} \forall \, E \in \mathcal{F} : X \setminus E \in \mathcal{F} \end{equation} |
| (3) | \begin{equation} \forall \, E_0, E_1, E_2, \dots \in \mathcal{F} : \bigcup_{n \in \mathbb{N}} E_n \in \mathcal{F} \end{equation} |
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Comment 0
Occasionally, one might see condition (1) requiring that both $\emptyset$ and $X$ are in $\mathcal{F}$. However, results R2067: Subtracting empty set from set and R2066: Difference of set with itself show that $X \setminus \emptyset = X$ and $X \setminus X = \emptyset$, so that condition (2) ensures $\emptyset, X \in \mathcal{F}$ if only one of $\emptyset$ or $X$ is assumed to belong to $\mathcal{F}$.
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| ▶ | D1916: Bottom sigma-algebra |
| ▶ | D195: Discrete sigma-algebra |
| ▶ | D1729: Pushforward sigma-algebra |
| ▶ | D484: Set of sigma-algebras |
| ▶ | D470: Subsigma-algebra |
