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

  • $\begingroup$ 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

$$x\in \bigcup_{i\in I}A_i\iff \exists i\in I,\ x\in A_i \\ x\in \bigcap_{i\in I}A_i\iff \forall i\in I,\ x\in A_i$$

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.