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Let $(K_n)$ be a sequence of sets.

What is the negation of the following statement?

For all $U$ open containing $x$, $U \cap K_n \neq \emptyset$ for all but finitely many $n$.

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In my opinion, for homework-type questions like this, we should not give a complete answer immediately. Instead we whould ask the OP for his thoughts and attempts. However: Asaf and Davide have far higher reputations than I, so I guess they know best... – GEdgar Sep 21 '12 at 15:49
@GEdgar: I suppose it is a matter of opinion, experience and approach. I prefer to give complete and detailed solutions sometimes because I think they are valuable more than just giving hints. I also don't see how this is properly a homework question. It might be something remotely connected to a homework question but tagged general topology I doubt that is the entire question. – Asaf Karagila Sep 21 '12 at 15:54
This was not the entire question, by far. It actually didn't come from a question, but rather, a couple remarks about the definition of an upper topological limit (and lower). It's just that I was having trouble writing it out logically and getting the negation. I spoke to some other students here and they were having trouble also. – user41728 Sep 21 '12 at 20:38

3 Answers 3

up vote 7 down vote accepted

Let us write the statement formally: $$\forall U(x\in U\rightarrow\exists n\forall k(k>n\rightarrow U\cap K_k\neq\varnothing))$$ For every $U$ (open of course), if $x\in U$ then there is some $n$ that for all $k>n$ we have $K_k\cap U\neq\varnothing$.

Now negation flips quantifiers and $\lnot(\alpha\rightarrow\beta)$ is the same as $\lnot\beta\land\alpha$. So we have:

$$\exists U(x\in U\land\forall n\exists k(k>n\land U\cap K_k=\varnothing))$$ Or in words, there exists an open set $U$ such that $x\in U$ but for every $n$ there is some $k>n$ such that $U\cap K_k=\varnothing$. However in the natural numbers to say that something happens unboundedly often is the same as saying it happens infinitely often. So finally we can say:

There exists an open set $U$ such that $x\in U$ and for infinitely many $n$ we have $U\cap K_n=\varnothing$.

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Exists an open set containing $x$ such that $U\cap K_n$ is empty for infinitely many $n$.

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$\neg$(for all $U$ open containing $x$, $U \cap K_n \ne \emptyset$ for all but finitely many $n$).

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