# Denumerable union/intersection of empty sets

Le $$X$$ be a topological space.

Consider the empty set $$\emptyset$$. I have no problem with the facts that $$\emptyset\cup\emptyset=\emptyset$$ and that $$\emptyset\cap\emptyset=\emptyset$$.

But what if I have a denumerable union or a denumerable intersection of empty sets. Do I have $$\emptyset\cup\emptyset\cup\dots=\emptyset$$ and $$\emptyset\cap\emptyset\cap\dots=\emptyset$$?

(I am concerned because the empty set is an open-closed set, but denumerable unions and intersections of clopen sets are not clopen, so I might rather have $$\emptyset\cup\emptyset\cup\dots\subseteq\emptyset$$ and $$\emptyset\cap\emptyset\cap\dots\supseteq\emptyset$$.)

• Denumerable intersections of clopen sets are not necessarily clopen. But sometimes they are, as in this case. – Alex Kruckman May 25 at 16:24

$$x\in \bigcup_{i\in I}A_i\iff \exists i\in I,\ x\in A_i \\ x\in \bigcap_{i\in I}A_i\iff \forall i\in I,\ x\in A_i$$
If $$I\ne\emptyset$$ and $$A_i=\emptyset$$ for all $$i\in I$$, then $$x\in A_i$$ is false for all $$i$$, therefore both $$\exists i\in I, x\in A_i$$ and $$\forall i\in I, x\in A_i$$ are false.