# ultrafilter convergence versus non-standard topology

I have recently been reading about the non-standard characterisation of topological spaces, by saying which points of ${^*X}$ are infinitesimally close to which standard points. The theory looks a lot like that of ultrafilter convergence. Let me clarify:

• A point $x$ lies in the closure of $A$ iff there is an ultrafilter $\mathcal{U}$ on $A$ with $\mathcal{U}\to x$
• A point $x$ lies in the closure of $A$ iff there is a non-standard $y \in {^*A}$ with $y\simeq x$

Here, ultrafilters play the same role as non-standard points, and convergence plays the same role as the $\simeq$ relation. Another example:

• X is Hausdorff iff for every ultrafilter $\mathcal{U}$ we have $(\mathcal{U}\to x \mbox{ and } \mathcal{U}\to y) \implies x=y$
• X is Hausdorff iff for every non-standard $z$ we have $(z\simeq x \mbox{ and } z\simeq y) \implies x=y$

Again we just have to change ultrafilters to non-standard points and convergence to $\simeq$ to get the non-standard version of the property. This analogy also works for compactness and products, yielding the same easy proof of Tychonov.

We have the following correspondence (let $A\subseteq X, x\in X$): \begin{aligned} y\in {^*X} &\leftrightarrow \mathcal{U} \mbox{ ultrafilter on } X\\ y \simeq x &\leftrightarrow \mathcal{U} \to x\\ y\in {^*A} &\leftrightarrow \mathcal{U} \mbox{ ultrafilter on } A \mbox{ (i.e. containing A)}\\ ^*A &\leftrightarrow \{\mathcal{U}\mbox{ ultrafilter } \mid A \in \mathcal{U} \}\\ ^*x &\leftrightarrow \mbox{ the principal filter on x} \end{aligned}

My first question is, how far does this analogy go? Is the ultrafilter characterisation a special case of non-standard topology? Can we then also use it to describe the star of functions, relations, ... , i.e. does the above correspondence induce a monomorphism?

My second question is, if we apply this analogy to the space $({^*X}, \mathcal{T}_S)$ carrying the standard topology (which is compact if the extention is saturated), then we get the space of all ultrafilters on $X$ with basis open sets $\{\mathcal{U} \mid G \in \mathcal{U} \}$ for all opens sets $G \subseteq X$. Is this space compact (like the analoguous space in the non-standard setting)? Does this construction have a name, in the ultrafilter setting? According to this analogy, its Hausdorff reflection should be the Stone-Cech compactification $\beta X$. This does agree with the ultrafilterconstruction of $\beta X$ for discrete $X$

Remark: It is very difficult to google this question because ultrafilters (on a different set) are used to construct non-standard extentions, so any search containing ultrafilters and non-standard analysis leads me to that construction and not to the use of ultrafilter convergence to describe topology. A link to a book or paper on this topic, or even a title would be a satisfactory answer.

• Have you heard of the Wallmann Extension? – DanielWainfleet May 15 '18 at 13:26
• @DanielWainfleet that sounds interesting. Can you comment more fully? – Mikhail Katz May 16 '18 at 8:47
• @MikhailKatz . See General Topology by R. Engelking. For a completely regular space $X$ there's is a dense homeomorphic embedding $w:X\to wX$ where $wX$ is compact and any continuous $f:X\to Y$ (with $Y$ compact Hausdorff) extends to a continuous $f^*:wX\to Y.$... If $X$ is normal, $wX$ is equivalent to $\beta X.$ If $X$ is not normal, $wX$ is not a $T_1$ space....Let $C$ be the set of closed subsets of $X.$ Let $F \subset C$ such that (i) $\phi \not \in F \ne \phi$, (ii) $A,B\in F\implies A\cap B\in F$, (iii) $(A\subset B\in C \land A\in F) \implies B\in F$, (continued) – DanielWainfleet May 17 '18 at 14:33
• ...(continued)...and (iv) $F$ is subset-maximal among subsets of $C$ with properties (i),(ii),(iii). Then $wX$ is the set of all such $F$.... For $p\in X$ let $w(p)=\{c\in C:p\in C\}$.... Let $O$ be the set of open subsets of $X$. A base for the topology on $wX$ is $\{U^*: U\in O\}$ where $U^*=\{F\in wX: \exists c\in F\;(c\subset U\}$....Discrete spaces are normal. If $X$ is discrete then $wX$ is the set of all ultrafilters on $X$ and for $p\in X,\;$ $w(p)$ is the principal ultralfilter $\{A\subset X:p\in A\},$ and for $U\subset X,\;$ $U^*=\{F\in wX:U\in F\$. – DanielWainfleet May 17 '18 at 14:48
• Engelking means "compact Hausdorff" when he writes "compact". He calls a space with the "compactness property" but not necessarily Hausdorff a "pseudo-compact" space..... Proving that $wX$ has the compactness property is tricky but not difficult/ – DanielWainfleet May 17 '18 at 14:54

The existence of such a relation is not surprising because a nonstandard number $x\in{}^{\ast}\mathbb R$ can be used to specify an ultrafilter. If $x\in{}^{\ast}\mathbb N\setminus \mathbb N$ then the corresponding ultrafilter is $\{A\in\mathcal{P}(\mathbb N)\colon {}^{\ast}\!A\ni x\}$. If $x\not\in{}^\ast\mathbb N$ argue similarly. For instance if $x$ is infinite, use instead $\lfloor x \rfloor\in{}^\ast\mathbb N$. If $x$ is finite use $\frac{1}{x-\mathbf{st}(x)}$, etc.