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I have come across the term "ultra prefilter" which has two possible definitions (that I can think of). I tried googling this first I swear! (but google thinks I'm looking for filtered water or a water purifier)

The two most obvious meanings to assign to an ultra prefilter $F$ on a space $X$

  • $F$ is a prefilter on $X$ which is not properly contained in any other prefilter on $X$

  • $F$ is a prefilter on $X$ which is equivalent to any of its refinements

The first one is simply the definition of an ultra filter applied to a prefilter instead. However the second one seems more appropriate, but I wanted to confirm it before I trust it too deeply.

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An ultra prefilter is a prefilter, such that all supersets of the prefilter form an ultrafilter. That is, it is a filter base of an ultrafilter. You can take a look at 5.2. in Pete L. Clark's notes on convergence.

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Those are indeed the notes I am working through, but I couldn't find a definition for ultra prefilter. As for your answer, I understand everything after the "That is," part. But before that, can you clarify exactly what needs to form a filter? The set of all supersets of sets in the prefilter always forms a filter I think. –  Kyle Schlitt May 23 '12 at 22:47
1  
@Kyle: That was a typo. They are supposed to form an *ultra*filter. The terminology is not standard. The definition of a prefilter comes right after Proposition 5.5. on page 19. The conventions for how to relate filter-concepts and prefilter-concepts is on page 20. –  Michael Greinecker May 23 '12 at 22:51
    
It's funny, I know exactly what paragraph you are referring to when you mention the "conventions", I remember the paragraph exactly, yet I didn't make the connection..... Thanks for the answer! –  Kyle Schlitt May 23 '12 at 22:57
    
You are welcome! –  Michael Greinecker May 23 '12 at 22:59

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