What is a z-filter?

Question

I am trying to understand meaning of z-filters (def. below).

Why are they introduced?

Where are they used in mathematics? The only example I have found is that they are used for the construction of Stone-Čech compactification of non-discrete topological space (but I dont know how and appreciate answer to this too).

How does the definition of z-filters connect with the definition of filters? What can we tell about z-filters?

Thank you for any advice or resources for z-filters.

Definition

A nonempty subfamily $F$ of $Z(X)$ is called $z$-filter on $X$ provided that

1. $∅∉F$
2. If $z_1,z_2∈F$ , then $z_1∩z_2∈F$
3. If $z∈F,z^∗∈Z(X),z^∗⊃z$ , then z^∗∈F

The family $Z[C(X)]=Z(X)=\{Z(f):f∈C(X)\}$ is all zero-sets in X.

$Z(f)=\{x∈X:f(x)=0\}$

Answer 1

I explained filters in posets with meets in my answer here. A $z$-filter is just a special case for the case of $(Z(X), \subseteq, \cap)$, where $Z(X) = \{Z(f) \mid f \in C(X)\}$, $C(X)$ is the ring of continuous real-valued functions on $X$ and $Z(f)=f^{-1}[\{0\}]$. It's just a special family of closed subsets of $X$ and as such a meet-semilattice and so the definition in the other answer applies in particular.

They are mostly introduced because it gives a connection between $\beta X$ and the ring of continuous functions $C(X)$. When $I$ is an ideal in that ring, one considers the set $Z[I] = \{Z(f): f \in I\}$ and it turns out this collection is a $z$-filter. This gives a nice correspondence between ideals in the ring and what turns out to be the points in $\beta X$. A classic text on this is Rings of Continuous Functions by Gillman and Jerrison.

Answer 2

Do you know the more ordinary notion of "filter"? If not, becoming familiar with that first will help.

I know about $z$-filters from the great book

Gillman, Leonard; Jerison, Meyer, Rings of continuous functions., Mineola, NY: Dover Publications (ISBN 978-0-486-81688-3). ix, 300 p. (2017). ZBL1380.46022.

$z$-filters are used in this part of mathematics: where real analysis meets point-set topology.