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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 $\emptyset\cap\emptyset\cap\dots\supseteq\emptyset$.)

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  • $\begingroup$ Denumerable intersections of clopen sets are not necessarily clopen. But sometimes they are, as in this case. $\endgroup$ – Alex Kruckman May 25 at 16:24
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$$x\in \bigcup_{i\in I}A_i\iff \exists i\in I,\ x\in A_i \\ x\in \bigcap_{i\in I}A_i\iff \forall i\in I,\ x\in A_i$$

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.

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