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I'm on chapter 5 now in these notes: http://math.uga.edu/~pete/convergence.pdf

I'm stuck trying to prove Proposition 5.6. (on the top of page 21/bottom of page 20).

First, I think that the first $\mathcal F$ that appears in the statement of the proposition should really be a $F$ (where previously $F$ was used for a pre-filter, and then $\mathcal F$ for the associated filter which is generated by taking all supersets of sets which are contained in $F$). So I have been working under this assumption.

My interpretation is that I have two pre-filters on a topological space $X$: $F$ and $F'$, and the two respective associated filters: $\mathcal F$ and $\mathcal F'$.

The statement that I cannot prove is the $(\Leftarrow)$ direction for part (b).

That is, I cannot prove the following statement:

If $\mathcal F'$ converges to $x$, then $F$ converges to $x$.

If anyone has any suggestions on how to prove this I would be very grateful. If necessary I can supply my arguments for the other parts.

Thanks as always!

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    That being said, it is worth noting the part I *was* asking about was reproduced in full. I still feel that my question was completely self-contained. Anyways, I do not wish to be argumentative (though I probably already have been here). Please trust that I will think extensively about this when posting any further questions.2012-05-24

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There are several typos. You’re right about the first $\mathcal F$ in the statement of the proposition: it should indeed be $F$. Clause (b) of the proposition should read:

b) $F$ converges to $x\iff\mathcal{F}$ converges to $x$.

The true half of the implication that’s actually written for (b) appears in its proper place as (d).

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I have uploaded a corrected version of the document.

Thanks again for your close reading.

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    You might want to take a look at the proof of Theorem 3.6; the gap noticed [here](http://math.stackexchange.com/questions/144099/proof-of-kelleys-theorem) is still there.2012-05-25