Zero set filter and zero set ultrafilter

Let $X$ be a Tychonoff space (Completely regular spaces and Tychonoff ($T_{3\frac{1}{2}}$) spaces) and let $\mathbb{R}$ carry its usual topology. Put

$C(X):=C(X,\mathbb{R}),$

the ring of all continuous real-valued functions on $X$ with pointwise addition and multiplication. No boundedness and no norm is assumed: functions in $C(X)$ may be unbounded, and $C(X)$ is not treated as a Banach algebra anywhere on this page. For $f\in C(X)$ let

$Z(f):=f^{-1}[\{0\}]=\{x\in X:f(x)=0\}$

be the zero set of $f$ (Zero sets and cozero sets of continuous real-valued functions), and call a subset of $X$ a zero set of $X$ when it is $Z(f)$ for some $f\in C(X)$. Write $\mathcal{Z}(X)$ for the family of all zero sets of $X$.

A z-filter on $X$ is a family $\mathcal{F}\subseteq \mathcal{Z}(X)$ of zero sets with

1. $X\in \mathcal{F}$;
2. $\mathrm{\varnothing }\mathrm{\notin }\mathcal{F}$;
3. $\mathcal{F}$ is closed under finite intersections: if $Z,Z^{\mathrm{\prime }}\in \mathcal{F}$ then $Z\cap Z^{\mathrm{\prime }}\in \mathcal{F}$;
4. $\mathcal{F}$ is upward closed inside $\mathcal{Z}(X)$: if $Z\in \mathcal{F}$ and $Z\subseteq Z^{\mathrm{\prime }}$ with $Z^{\mathrm{\prime }}\in \mathcal{Z}(X)$, then $Z^{\mathrm{\prime }}\in \mathcal{F}$.

A z-ultrafilter on $X$ is a z-filter that is maximal among z-filters with respect to inclusion: a z-filter $\mathcal{U}$ such that every z-filter $\mathcal{G}\supseteq \mathcal{U}$ satisfies $\mathcal{G}=\mathcal{U}$.

Two elementary facts about $\mathcal{Z}(X)$ are used repeatedly and are recorded here rather than reproved each time:

• $\mathcal{Z}(X)$ is closed under finite intersections, because $Z(f)\cap Z(g)=Z(f^2+g^2)$ for all $f,g\in C(X)$; in particular $Z(f)\cap Z(g)$ is again a zero set and the condition 3 of a z-filter is not vacuous. Similarly $X=Z(0)$ and $\mathrm{\varnothing }=Z(1)$ are zero sets, so the conditions 1 and 2 are meaningful.
• If $Z(f)\subseteq Z(g)$, no algebraic formula for $g$ in terms of $f$ is claimed; inclusions of zero sets are handled through maximal ideals in Maximal ideals of C(X) and zero set ultrafilters.

• Why zero sets and not arbitrary closed sets. Arbitrary closed sets are also closed under finite intersections. What is special here is that the intersection remains represented by continuous functions through the explicit identity $Z(f)\cap Z(g)=Z(f^2+g^2)$; this function-theoretic representation is what connects z-filters to ideals of $C(X)$.
• Source status. The historical target (the neighbouring deferral is recorded as a remark on the companion examples page) was inaccessible in this run, and the failed recovery record is in the Batch 4 coverage ledger. This definition and its consumers (Maximal ideals of C(X) and zero set ultrafilters, Zero set ultrafilters and Stone-Cech points, Gelfand-Kolmogorov for rings of continuous functions) are complete local proofs, not source citations.