Zero set ultrafilters and Stone-Cech points

For $p\in \beta X$ define ${\rho }_p(u):=\bar{u}(p)$ for $u\in C(X,[0,1])$; then $\mathrm{\mid }{\rho }_p(u)\mathrm{\mid }\le 1$ and ${\rho }_p$ is additive, positively homogeneous, multiplicative and lattice-preserving: for $u,v\in C(X,[0,1])$ and real $\lambda ,\mu \ge 0$ with $\lambda u+\mu v$ again $[0,1]$-valued, ${\rho }_p(\lambda u+\mu v)=\lambda {\rho }_p(u)+\mu {\rho }_p(v)$, ${\rho }_p(uv)={\rho }_p(u){\rho }_p(v)$, and ${\rho }_p(\max (u,v))=\max ({\rho }_p(u),{\rho }_p(v))$; more generally every polynomial identity with nonnegative coefficients valid on $X$ passes to ${\rho }_p$.

If $(x_{\alpha })$ is a net in $X$ with $e(x_{\alpha })\to p$ in $\beta X$, then ${\rho }_p(u)={\lim }_{\alpha }u(x_{\alpha })$ for every $u\in C(X,[0,1])$: this is continuity of $\bar{u}$ at $p$ together with $\bar{u}\circ e=u$.

Characterisation of closure points. For $A\subseteq X$ one has $p\in {\overline{e[A]}}^{\beta X}$ if and only if ${\rho }_p(u)=0$ for every $u\in C(X,[0,1])$ with $u{\mathrm{\mid }}_A=0$. If $p\in \overline{e[A]}$ choose a net $a_{\alpha }\in A$ with $e(a_{\alpha })\to p$ and use [step 1.2]. Conversely, if $p\mathrm{\notin }\overline{e[A]}$ take a basic neighbourhood $N=\{q:\mathrm{\mid }q(u_i)-{\rho }_p(u_i)\mathrm{\mid }<\epsilon ,i\le n\}$ of $p$ in the cube with $N\cap e[A]=\mathrm{\varnothing }$ and put $w:=\max (0,1-{\epsilon }^{-2}{\sum }_{i\le n}(u_i-{\rho }_p(u_i){)}^2)\in C(X,[0,1])$; then ${\rho }_p(w)=\max (0,1-0)=1$ by [step 1.1], while $w(a)=0$ for every $a\in A$, since $e(a)\mathrm{\notin }N$ means ${\sum }_i(u_i(a)-{\rho }_p(u_i){)}^2\ge {\epsilon }^2$.

For $p\in \beta X$, the family ${\mathcal{U}}_p$ is a z-filter: it contains $X$ because $e[X]$ is dense in $\beta X$; it omits $\mathrm{\varnothing }$ because ${\rho }_p(1)=1\mathrm{\ne }0$ and $1$ vanishes on $\mathrm{\varnothing }$; it is upward closed because $Z\subseteq Z^{\mathrm{\prime }}$ implies $\overline{e[Z]}\subseteq \overline{e[Z^{\mathrm{\prime }}]}$; and it is closed under finite intersections: if $Z_i=Z(f_i)$ with $p\in \overline{e[Z_i]}$ for $i=1,2$, take $u\in C(X,[0,1])$ vanishing on $Z_1\cap Z_2$ and write $u=u_1+u_2$ with $u_1,u_2\in C(X,[0,1])$ vanishing on $Z_1$, respectively $Z_2$, by the construction of [L4]; then ${\rho }_p(u_i)=0$ by [step 2.1] and ${\rho }_p(u)=0$ by [step 1.1], so $p\in \overline{e[Z_1\cap Z_2]}$ by [step 2.1] again.

Injectivity. For $u\in C(X,[0,1])$ and real $c$ one has $p\in \overline{e[\{u\le c\}]}$ whenever $c>{\rho }_p(u)$: for a net $x_{\alpha }$ with $e(x_{\alpha })\to p$ one has $u(x_{\alpha })\to {\rho }_p(u)<c$ by [step 1.2], so eventually $x_{\alpha }\in \{u\le c\}$. Conversely, if $c<{\rho }_p(u)$ then $p\mathrm{\notin }\overline{e[\{u\le c\}]}$: with $v:=\min (1,(u-c{)}^{+})$ one has ${\rho }_p(v)=\min (1,{\rho }_p(u)-c)>0$ by [step 1.1] and $v$ vanishes on $\{u\le c\}$, so [step 2.1] applies. Hence ${\rho }_p(u)=\inf \{c\in \mathbb{R}:Z((u-c{)}^{+})\in {\mathcal{U}}_p\}$ depends only on ${\mathcal{U}}_p$, and since the coordinates ${\rho }_p(u)$ over $u\in C(X,[0,1])$ determine the point $p$ of the cube by [L2], the equality ${\mathcal{U}}_p={\mathcal{U}}_q$ forces $p=q$.

For $p\in \beta X$ the z-filter ${\mathcal{U}}_p$ is maximal. Let $\mathcal{W}\supseteq {\mathcal{U}}_p$ be a z-filter and let $Z\in \mathcal{Z}(X)$ with $Z\in \mathcal{W}$; if $Z\mathrm{\notin }{\mathcal{U}}_p$, then $p\mathrm{\notin }\overline{e[Z]}$ and [step 2.1] provides $u\in C(X,[0,1])$ vanishing on $Z$ with $t:={\rho }_p(u)>0$; the zero set $Z_1:=Z((t\mathrm{/}2-u{)}^{+})$ is disjoint from $Z$, because $u=0$ on $Z$ makes $(t\mathrm{/}2-u{)}^{+}=t\mathrm{/}2$ there, and belongs to ${\mathcal{U}}_p$, because for any net $x_{\alpha }\in X$ with $e(x_{\alpha })\to p$ one has $u(x_{\alpha })\to t>t\mathrm{/}2$ by [step 1.2], so eventually $(t\mathrm{/}2-u(x_{\alpha }){)}^{+}=0$, that is, $x_{\alpha }\in Z_1$, whence $p\in \overline{e[Z_1]}$; but then $Z,Z_1\in \mathcal{W}$ give $\mathrm{\varnothing }=Z\cap Z_1\in \mathcal{W}$, contradicting that $\mathcal{W}$ is a z-filter. Hence $Z\in {\mathcal{U}}_p$, and ${\mathcal{U}}_p$ is a z-ultrafilter.

Surjectivity. Let $\mathcal{U}$ be a z-ultrafilter. The family $\{\overline{e[Z]}:Z\in \mathcal{U}\}$ consists of closed subsets of the compact space $\beta X$ and has the finite intersection property, because the intersection of finitely many such closures contains $\overline{e[Z_1\cap \cdots \cap Z_n]}$ with $Z_1\cap \cdots \cap Z_n\in \mathcal{U}$ nonempty; by [L3] there is $p$ in the intersection, so $p\in \overline{e[Z]}$ for every $Z\in \mathcal{U}$, that is, $\mathcal{U}\subseteq {\mathcal{U}}_p$; both are z-filters and $\mathcal{U}$ is maximal, so $\mathcal{U}={\mathcal{U}}_p$ by [step 4.1].

By [step 4.1] every ${\mathcal{U}}_p$ is a z-ultrafilter, by [step 5.1] the map $p\mapsto {\mathcal{U}}_p$ is surjective, and by [step 3.2] it is injective; hence it is a bijection onto the set of z-ultrafilters.