Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 2 down vote accepted

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.

share|cite|improve this answer
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. – roo May 23 '12 at 22:47
@Kyle: That was a typo. They are supposed to form an ultrafilter. 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! – roo May 23 '12 at 22:57
You are welcome! – Michael Greinecker May 23 '12 at 22:59

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.