# Eternal preliminaries part 2, filters and ultrafilters

Last time I wrote about the basic structures, (partial) semigroups. But algebra in the Stone–Čech compactification deals, well, with the Stone–Čech compactification. I will try to ignore the general theory of compactifications because we only deal with a very simple case — discrete spaces. Suffice it to say that any elementary topology book should have a chapter on compactifications if you want to read more.

## Filters and ultrafilters

Already at the end of the first part, I needed to refer to the notions of filters. I don’t want to talk too much about filters or ultrafilters formally because a) we’re going to talk about them all the time anyway and b) wikipedia (to which I link) is much better at giving you a concise but broad overview than I am. Let me give you a cheat sheet though.

- A family of subsets of
is a **filter**if it does not contain, is closed under taking finite intersections and supersets. - A family of subsets of
has the **(strong or infinite) finite intersection property or (i)FIP**if the intersection of any finite subfamily is non-empty (infinite). - A family with (i)FIP generates a (free) filter by closing it under finite intersections and supersets.
- A filter
is an ultrafilter iff it is a **maximal**filter (with respect to inclusion) iff it is**prime**iff . - We identify
with the principal ultrafilter .

Filters can be considered as 0-1-valued, finitely-additive measures (or rather the measure 1 sets of such a measure) in which case ultrafilters are complete measures which is an idea I might use in “prose” every once in a while. You can also consider them as a form of universal quantifiers which gives another intuition. A useful shorthand will be “almost all (with respect to some ultrafilter

We’ll get back to this later. One final note,

**The Ultrafilter Lemma** Every filter can be extended to an ultrafilter.

**Proof**.

- The set of filters is partially ordered by inclusion.
- The union of any chain of filters is a filter.
- Apply Zorn’s Lemma to find a maximal element.

This theorem is weaker than the axiom of choice, but very strong already in itself (looking up that link I just learned that Tarski himself proved the existence of non-principal ultrafilters in 1930; wikipedia. awesome.). Of course, the real power lies in the three characterizations of ultrafilters in the cheat sheet, so let’s prove the difficult one

A filter

**Proof**.

- If
is maximal, , then either or . - If there exists
such that , then , so and we’re done. - Otherwise every
has in which case has the FIP, hence generates a filter. - But
was maximal, so the generated filter cannot gain new elements. - In other words,
.

- If there exists
- If
is prime, then is maximal. - Consider any family
such that , but there exists . - Since
, then by primeness of , . - Therefore,
are both in . - In other words,
is not a filter — in other words, is maximal.

- Consider any family

## The Stone–Čech compactification (apologies for missing haceks)

The set of ultrafilters is often denoted by

**Universal Property of ** If

The easiest way to do this in our setting, is to take the limit along ultrafilters. But for now we don’t need to.

**An interlude about extensions** If

Often this definition is given by

## Extending the semigroup operation

We want to extend our (partial) semigroup operation to

I “grew up” with the book by Neil Hindman and Dona Strauss, so I tend to follow their set up (regarding which kind of extension we want).

**Using the universal property of **

- For each
, we can consider , i.e., multiplication with a fixed left-hand side. - This is a continuous map (since any map on
is), so we can extend it to ; we simply write for this. - Now switch it around and for
consider this extended multiplication with fixed on the right hand side, i.e., the map . - Again this is a continuous map (since any map on
is), so we can extend it to ; and for this we write . - Tada, we have our operation.

Of course, this gives us no tangible clue as to what such a product of ultrafilters actually looks like. But at least one thing is easy — multiplication with a fixed right hand side is continuous! I call this **right-topological**. You can see that we might start symmetrically and then we end up with a different operation (though very similar to our own). Also, some people like to call the above continuity left-topological (because its continuous in the left hand argument). So, lots of confusion… we’ll stick to this one.

**The brute force definition**

There’s thankfully a way to give the same definition by brute force (which is my favourite way to write it down), but let’s think about it naively. We have two ultrafilters

So what do we need to do? We need to complicate things (and if you try to write down to check where the above attempt of a definition fails, this complication comes naturally). Later I’ll introduce some notation to make nicer general nonsense, but let’s take a look first.

**Extending multiplication to ** For a semigroup

Ok, quite a beast. Don’t despair! Remember what we tried first: sets of the form

There is a different angle to look at this: the tensor (or Fubini) product of ultrafilters.

**Tensor product of ultrafilters** For

Not much better, eh? Let’s take a look though: the tensor product is contains sets

What has this to do with the product we defined before? Well, the tensor product live on

Still not happy? Yeah, I know that feeling… Ok, let me offer my favourite general nonsense notation.

- For
define (note: don’t have to be able to invert to define this…) - For
define - Then
if and only if .

Alright, much shorter now. But does it help? I don’t know. I certainly don’t claim to “really” understand this operation (but there’s a certain limit since, well, it’s on ultrafilters after all…). My notation for the set

One advantage is that you can check a few things more easily with the brute force definition.

- The operation is right-continuous —
is exactly the neighbourhood that shows the continuity of with respect to . - The operation is associative — just write it out.

Phew, that was a lot. But we’re finally ready to get to some real theorems!