Why define $\bigcup_{n\in I} A_n=\emptyset,\bigcap_{n\in I} A_n=\Omega$ when $I$ is the empty set
36881
Empty intersection and empty union
46240
Show that if $A$ and $B$ are sets, then $(A\cap B) \cup (A\cap \overline{B})=A$.
126821
Let $f:[0,1]\to \mathbb{R}$. Show that $\{a \in [0,1]: f(a)\neq 0\} = \bigcup_{n=1}^{\infty} \{a \in [0,1]: |f(a)|\geq 1/n\}$
23442
Cardinality of the union of infinite and countable sets
132868
Bijection between an infinite set and its union of a countably infinite set
62208
If $A$ is a non-empty set, then $A \notin A$
31212
The existence of the empty set is an axiom of ZFC or not?
88823
How can the empty set be open in the definition of the cofinite topology?
18828
Base and empty set of a topology
148927
Why does every countable limit ordinal have cofinality $\omega$?
150825
Is an empty set equal to another empty set?
148213
Given a topological space $X$, why are both $X$ and $\emptyset$ open and closed?
98816
How to prove that $\{f+g<a\}=\bigcup_{r\in{\Bbb Q}}(\{f<r\}\cap\{g<a-r\})$?
68230
The cardinality of a countable union of sets with less than continuum cardinality
78763
Prove that $ S=\{0\}\cup\left(\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}\right)$ is a compact set in $\mathbb{R}$.
127720
Prove that $\bigcup_{n \in \Bbb{N}} [1/n,1] = (0,1]$.
138069
Proof that $\bigcap_{a\in A} \bigg(\, \bigcup_{b\in B} F_{a,b} \, \bigg) = \bigcup_{f\in ^AB} \bigg(\, \bigcap_{a \in A} F_{a,f(a)}\,\bigg) $