In topology, I read that sometimes we can add the empty set to a set that is not a topology to make it a topology, I'm ok with this, but my question is, how there could be a set that doesn't contain the empty set; is it not that this is the empty set {} and every set whatever in it, from the empty set itself to infinite sets (or any other arbitrary sets) has at least this empty container i.e. the empty set?
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1$\begingroup$ What do you mean by "contain"? Does a set contain its elements, or does it contain its subsets? $\endgroup$– bofSep 25, 2015 at 9:34
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$\begingroup$ the power set of any set contains the empty set, if that's what you are looking for $\endgroup$– user265328Sep 25, 2015 at 9:34
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1 Answer
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If by "$A$ contains $B$", you mean "$B$ is a subset of $A$", then the answer is no. Because the statement
$\forall x\in\{\}: x\in A$
Is always true, it is always true (no matter what $A$ is), that $\{\}\subseteq A$.
If by "$A$ contains $B$", you mean "$B$ is an element of $A$", then the answer is yes. For example, the empty set is not an element of the empty set. Also, it is not an element of $\mathbb N$, or $\mathbb C$, or $\{1,2,3\}$ or a lot of other sets.
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$\begingroup$ yes my problem was messing up "element" with "subset". $\endgroup$ Sep 26, 2015 at 16:16