# Verifying Properties of Filters

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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While including the link to the notes, please do your best to make questions self-contained. Include the content of the said proposition. –  Asaf Karagila May 23 '12 at 20:42
Please forgive me if I come across negatively here, I only am saying this because this issue has been brought up regarding my questions in the past. In this case I think the question is self-contained (aside from including definitions), despite the fact that it refers to the source from which the question came. I had however only extracted the part of the problem that I was having trouble with. –  roo May 23 '12 at 21:50
No, a self-contained question would have a part quoting the proposition at hand. It is not always a one-click thing to open a .pdf file. –  Asaf Karagila May 23 '12 at 21:51
I'm not trying to argue, but I still believe my question statement meets the criteria you supply. "The statement that I cannot prove is the $(\Leftarrow)$ direction for part (b). That is, I cannot prove the following statement: If $F′$ converges to $x$, then $F$ converges to $x$." –  roo May 23 '12 at 21:58
I agree with you 100%. I should certainly make every effort to ensure that the person who is in no way obligated to help me but nonetheless is donating their time for my benefit has no unnecessary difficulty in understanding my question. –  roo May 23 '12 at 22:10

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.