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Given a collection $\mathcal A$ of sets and a large set $X$. What are $\bigcup_{S\in\mathcal A}S$ and $\bigcap_{S\in\mathcal A}S$ ?

The problem is if $\mathcal A$ is empty, what do $\bigcup_{S\in\mathcal A}S$ and $\bigcap_{S\in\mathcal A}S$ mean ?

Muknres's "topology" page 12, if $\mathcal A$ is empty, then $\bigcup_{S\in\mathcal A} S=\emptyset$ and $\bigcap_{S\in\mathcal A}S=X$.

I 'm puzzled about this. Can someone tell me why? Thanks a lot!

Michael Hardy
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2016
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  • There are probably several other duplicates, some might be more fitting than others. This was the first that came to mind. – Asaf Karagila Jan 16 '15 at 19:33

1 Answers1

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False implies true. (All englishmen taller that 4 meters are vegetarian.) For each $A$ in the empty set, is false, so that the intersection is the total set, $X$. By complementarity, you get the other result, but you could have it directly in the same fashion.

Olórin
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