SCIENTIFIC PROGRAMS AND ACTIVITIES

Seminars

Consider a sentence $\phi$ of the infinitary logic $L_{\omega_1, \omega}$. In 1970, Morley introduced the notion of a scattered sentence, and showed that if $\phi$ is scattered then the class $I(\phi)$ of isomorphism types of countable models of $\phi$ has cardinality at most $\aleph_1$, and if $\phi$ is not scattered then $I(\phi)$ has cardinality continuum. The absolute form of Vaught's conjecture for $\phi$ says that if $\phi$ is scattered then $I(\phi)$ is at most countable. Generalizing previous work of Ben Yaacov and the author, we introduce here the notion of a separable model of $\phi^R$, which is a separable continuous structure whose elements are random elements of a model of $\phi$. We say that $\phi^R$ has few separable models if every separable model of $\phi^R$ is uniquely characterized up to isomorphism by a function that assigns probabilities summing to one to countably many elements of $I(\phi)$. In a previous paper, Andrews and the author showed that if $\phi$ is a complete first order theory and $I(\phi)$ is at most countable then $\phi^R$ has few separable models. We show here that this result holds for all $\phi$, and that if $\phi^R$ has few separable models then $\phi$ is scattered. Hence if the absolute Vaught conjecture holds for $\phi$, then $\phi^R$ has few separable models if and only if $I(\phi)$ is countable, and also if and only if $\phi$ is scattered. Moreover, assuming Martin's axiom for $\aleph_1$, we show that if $\phi$ is scattered then $\phi^R$ has few separable models.

June 26, 2015

Franklin Tall, PFA(S)[S] II

This is a continuation of last week's lecture. Last week's lecture was largely motivation; this lecture will be mainly technical, developing the method. If you really want to attend and missed last week, contact me and I will give you something to read.

June 19, 2015

Franklin Tall, PFA(S)[S] and locally countable subspaces of compact countably tight spaces.

I have lectured many times in the seminar on Stevo’s method of forcing with a coherent Souslin tree S over a model of PFA restricted to posets that preserve S, since it has many interesting applications in set-theoretic topology. However I believe the current cohort of graduate students has not seen an actual proof of this sort. Since the seminar is suffering from a lack of speakers, I plan to give a sporadic series of lectures featuring such proofs. In particular, as soon as I understand it sufficiently well, I want to give Alan Dow’s proof that in such models, first countable perfect pre-images of omega_1 include copies of omega_1. This is the capstone of the proof of the consistency of every hereditarily normal manifold of dimension > 1 being metrizable. First of all, however, I want to prove a technical theorem that is necessary for the manifold result, and for many other results concerning under what conditions locally compact normal spaces are paracompact. This particular theorem – getting locally countable collections to be sigma-discrete - is perhaps not of wide interest, but the method of getting an uncountable set in such a model to be the union of countably many “nice” subsets (rather than just including an uncountable nice subset) should have more applications. The “proof” I gave of this result in the seminar five years ago turned out to have a gap. The gap is bridged by a clever idea of Stevo. The proof will appear in a joint paper.

June 12, 2015

Asger Törnquist, Definable maximal orthogonal families in forcing extensions

Two Borel probability measures nu and mu on Cantor space are orthogonal if there is a Borel set which has measure 1 for nu, but measure 0 for mu. An orthogonal family of measures is a family of pairwise orthogonal measures; it is maximal if it is maximal under inclusion.

Maximal orthogonal families of measures can't be analytic; this is a theorem of Preiss and Rataj (1985). A few years ago, Vera Fischer and I showed that in L there is a Pi-1-1 (lightface) maximal orthogonal family (a "mof") of measures in L, but that adding a Cohen real to L destroys all Pi-1-1 mofs. Subsequently, it was shown that the same holds if we add a random real (Friedman-Fischer-T.).

This motivated the question: Can a Pi-1-1 mof coexist with a non-constructible real? In this talk we answer this by showing there is a Pi-1-1 mof in the Sacks and Miller extensions of L. By contrast, we will see that in the Mathias extension of L there are no Pi-1-1 mofs, and in the process of doing so we will obtain a new proof of the Preiss-Rataj theorem.

This is joint work with David Schrittesser.

May 22, 2015

Francisco Kibedi, Maximal Saturated Linear Orders

In his 1907 paper about pantachies (maximal linearly ordered subsets of the space of real-valued sequences partially ordered by eventual domination), Felix Hausdorff poses several questions that he was unable to answer, including a question he labels $(\alpha)$: Is there a pantachie with no $(\omega_1, \omega_1)$-gaps?

Hausdorff knew that CH implies the answer is no; in other words, under CH, a pantachie must have $(\omega_1, \omega_1)$-gaps. However, Hausdorff's question turns out to be independent of ZFC. We answer question $(\alpha)$ by proving something a bit stronger, namely, Con(ZFC + $\lnot$CH + $\exists$ a maximal saturated linear order of size continuum in the space of real-valued sequences partially ordered by eventual domination). We then extend this result to include Martin's Axiom --- i.e., we prove Con(ZFC + MA + $\lnot$CH + $\exists$ a maximal saturated linear order of size continuum in the space of real-valued sequences partially ordered by eventual domination).

Note: This seminar will be held in the Bahen Center, room BA 1220.

May 1, 2015

Alessandro Vignati
Forcing axioms and Operator algebras: a lifting theorem for reduced products of matrix algebras

Inspired by the work of Farah and others in the application of forcing axioms to operator algebras, we prove a correspondent of a lifting theorem in a continuous setting. Analyzing different kinds of maps from the reduced product of matrix algebra into a corona of a nuclear C*-algebra, we provide different notions of well-behaved lifting, and we show how forcing axioms imply their existence, in contrast to the results obtained under the Continuum Hypothesis. Secondly, we show some consequences of such a behavior. All required definitions will be given. This is joint work with Paul McKenney.

April 17, 2015

Robert Raphael
On the countable lifting property for C(X)

Suppose that Y is a subspace of a Tychonoff space X so that the induced ring homomorphism $C(X) \rightarrow C(Y)$ is onto. We show that a countable set of pairwise orthogonal functions in C(Y) can be lifted to a pairwise orthogonal preimage in C(X). The question originally arose in vector lattices. Topping published the result for vector lattices using an erroneous induction, but two years later Conrad gave a counterexample. This is joint work with A.W.Hager.

April 10, 2015

Christopher Eagle
Model Theory of Compacta

Topological spaces do not fit well into the framework of first-order model theory; nevertheless, tools from model theory have had some success in applications to compacta. Model-theoretic ideas have been used in topology in two ways: First, by finding suitable first-order structures to use as stand-ins for topological spaces, and second, by directly "dualizing" notions from model theory. We will describe both of these methods, and compare them to a newer approach which applies real-valued logic to the rings of continuous complex-valued functions of compact spaces.

Using continuous logic we show that the pseudo-arc is a co-existentially closed continuum, answering a question of P. Bankston. We also show that the only compact metrizable spaces $X$ where $C(X)$ has quantifier elimination in continuous logic are the one-point space, the two-point space, and the Cantor set. This is joint work with Isaac Goldbring and Alessandro Vignati.

March 27, 2015

Dana Bartosova
About the conjecture that oligomorphic groups have metrizableuniversal minimal flows

We will discuss a conjecture of Lionel Nguyen van Th\'e as in the title. It was shown by Andy Zucker to be equivalent to whether every class of finitary approximations of a countable ultrahomogeneous structure with oligomorphic automorphism group has a finite Ramsey degree. We look at the problem from the Boolean algebra point of view. An interesting example in this context is the automorphism group of a topological structure whose natural quotient is the pseudo-arc, which is a work in progress with Aleksandra Kwiatkowksa (UCLA).

March 20, 2015

Room 230

Mike Pawliuk
Amenability and Directed Graphs Part 2 : Cherlin's List

Last week Miodrag spoke in general about Amenability, Fraisse classes and consistent random expansions. This talk will be more specific and focus on checking the amenability and unique ergodicity of the automorphism groups of the directed graphs on Cherlin's list. In addition, we will present a type of product of Fraisse classes that behaves nicely with respect to amenability and unique ergodicity.

Miodrag Sokic
Amenability and directed graphs

Amenability for locally compact and countable groups has been extensively studied. In this talk we will give some results in the case of non-archimedean groups. In particular, we consider groups of anthropomorphism of structures from the Cherlin list of ultrahomogeneous directed graphs.

Frank Tall
Some observations on the Baireness of C_k(X) for a locally compact space X

The area in-between Empty not having a winning strategy and Nonempty having a winning strategy in the Banach-Mazur game has attracted interest for many decades. We answer some questions Marion Scheepers asked when he was here last year, and also prove results related to his recent paper with Galvin and to a paper of Gruenhage and Ma. Our tools include PFA(S)[S] and non-reflecting stationary sets.

February 27, 2015

12:00pm

Saeed Ghasemi
Rigidity of corona algebras

In my thesis I use techniques from set theory and model theory to study the isomorphisms between certain classes of C*-algebras. In particular we look at the isomorphisms between corona algebras of direct sums of sequences of full matrix algebras. We will see that the question "whether any isomorphism between these C*-algebras is trivial" is independent from the usual axioms of set theory (ZFC). I also extend the classical Feferman-Vaught theorem to reduced products of metric structures. This theorem has a number of interesting consequences. In particular it implies that the reduced powers of elementarily equivalent structures are elementarily equivalent. We also use this to find examples of corona algebras of direct sums of sequences of full matrix algebras which are non-trivially isomorphic under the Continuum Hypothesis. This gives the first example of genuinely non-commutative structures with this property.

In the last chapter of my thesis I have shown that SAW*-algebras are not isomorphic to tensor products of two infinite dimensional C*-algebras, for any C*-tensor product. This answers a question of S. Wassermann who asked whether the Calkin algebra has this property.

February 11, 2015
3:30pm

Stewart Library

Alessandro Vignati
Set theory and amenable operator algebras

I will present my past and present work on logic and operator algebras. First I will show the construction of a nonseparable amenable operator algebra A with the property that every nonseparable subalgebra of A is not isomorphic to a C*-algebra, yet A is an inductive limit of algebras isomorphic to C*-algebras. Secondly, I will sketch possible techniques, associated to Model Theory in a continuous setting, that can be applied to operator algebras.

February 6, 2015

Room 230

Logan Hoehn
A complete classification of homogeneous plane compacta

In this topology talk, we will discuss homogeneous spaces in the plane $R^2$. A space X is homogeneous if for every pair of points in X, there is a homeomorphism of X to itself taking one point to the other. Kuratowski and Knaster asked in 1920 whether the circle is the only connected homogeneous compact space in the plane. Explorations of this problem fueled a significant amount of research in continuum theory, and among other things, led to the discovery of two new homogeneous spaces in the plane: the pseudo-arc and the circle of pseudo-arcs. I will describe our recent result which implies that there are no more undiscovered homogeneous compact spaces in the plane. This is joint work with Lex Oversteegen of the University of Alabama at Birmingham.

David Fernandez
A model of ZFC with strongly summable ultrafilters, small covering of meagre and large dominating number

Strongly summable ultrafilters are a variety of ultrafilters that relate with Hindman's finite sums theorem in a way that is somewhat analogous to that in which Ramsey ultrafilters relate to Ramsey's theorem. It is known that the existence of these ultrafilters cannot be proved in ZFC, however such an existencial statement follows from having the covering of meagre to equal the continuum. Furthermore, using ultraLaver forcing in a short finite support iteration, it is possible to get models with strongly summable ultrafilters and a small covering of meagre, and these models will also have small dominating number. Using this ultraLaver forcing in a countable support iteration to get a model with small covering meagre and strongly summable ultrafilters is considerably harder, but it can be done and in this talk I will explain how (it involves a characterisation of a certain kind of strongly summable ultrafilter in terms of games). Interesingly, this way we also get the dominating number equal to the continuum, unlike the previously described model.

Marcin Sabok
Automatic continuity for isometry groups

We present a general framework for automatic continuity results for groups of isometries of metric spaces. In particular, we prove automatic continuity property for the groups of isometries of the Urysohn space and the Urysohn sphere, i.e. that any homomorphism from either of these groups into a separable group is continuous. This answers a question of Melleray. As a consequence, we get that the group of isometries of the Urysohn space has unique Polish group topology and the group of isometries of the Urysohn sphere has unique separable group topology. Moreover, as an application of our framework we obtain new proofs of the automatic continuity property for the group $\mathrm{Aut}([0,1],\lambda)$, due to Ben Yaacov, Berenstein and Melleray and for the unitary group of the infinite-dimensional separable Hilbert space, due to Tsankov. The results and proofs are stated in the language of model theory for metric structures.

We show under $\mathfrak{p}=\mathfrak{c}$ that $P(\omega)/FIN$ can be embedded into the $P$-points under RK and Tukey reducibility.

November 28, 2014

12:30-3:00
Stewart Library

Antonio Avilés
A combinatorial lemma about cardinals $\aleph_n$ and its applications on Banach spa

The lemma mentioned in the title was used by Enflo and Rosenthal to show that the Banach space $L_p[0,1]^\Gamma$ does not have an unconditional basis when $|\Gamma|\geq \aleph_\omega$. In a joint work with Witold Marciszewski, we used some variation of it to show that there are no extension operators between balls of different radii in nonseparable Hilbert spaces.

Istvan Juhász
Lindelof spaces of small extent are $\omega$-resolvable

I intend to present the proof of the following result, joint with L. Soukup and Z. Szentmiklossy: Every regular space $X$ that satisfies $\Delta(X) > e(X)$ is $\omega$-resolvable, i.e. contains infinitely many pairwise disjoint dense subsets. Here $\Delta(X)$, the dispersion character of $X$, is the smallest size of a nonempty open set in $X$ and $e(X)$, the extent of $X$, is the supremum of the sizes of all closed-and-discrete subsets of $X$. In particular, regular Lindelof spaces of uncountable dispersion character are $\omega$-resolvable.

Miodrag Sokic
Functional classes

We consider the class of finite structures with functional symbols with respect to the Ramsey property.

Martino Lupini
Fraisse limits of operator spaces and the noncommutative Gurarij space

We realize the noncommutative Gurarij space introduced by Oikhberg as the Fraisse limit of the class of finite-dimensional 1-exact operator spaces. As a consequence we deduce that such a space is unique, homogeneous, universal among separable 1-exact operator spaces, and linearly isometric to the Gurarij Banach space.

Juris Steprans
The descriptive set theoretic complexity of the weakly almost periodic functions in the dual of the group algebra

The almost periodic functions on a group G are those functions F from G to the complex number such that the uniform norm closure of all shifts of F is compact in the uniform norm. The weakly almost periodic functions are those for which the analogous statement holds for the weak topology. The family of sets whose characteristic functions are weakly almost periodic forms a Boolean algebra. The question of when this family is a complete $\Pi^1_1$ set will be examined.

A cofinitary group is a subgroup of the group of all permutations of the natural numbers, all non-identity elements of which have only finitely many fixed points. A cofinitary group is maximal if it is not properly contained in any other cofinitary group. We will discuss the existence of nicely definable maximal cofinitary groups in the presence of large continuum and in particular, we will see the generic construction of a maximal cofinitary group with a $\Pi^1_2$ definable set of generators in the presence of $2^\omega=\aleph_2$.

Speaker 2 (from 13:30 to 15:00):
Menachem Magidor
On compactness for being $\lambda$ collectionwise hausdorff

A compactness property is the statement for a structure in a given class, if every smaller cardinality substructure has a certain property then the whole structure has this property. In this talk we shall deal with the compactness for the property of a topological space being collection wise Hausdorff. The space is X is said to be $\lambda$--collection wise Hausdorff ($\lambda$--cwH) if every closed discrete subset of X of cardinality less than $\lambda$ can be separated by a family of open sets. X is cwH if it is $\lambda$--cwH for every cardinal $\lambda$.

We shall deal with the problem of when $\lambda$--cwH implies cwH, or just when does $\lambda$--cwH implies $\lambda^+$--cwH. A classical example of Bing provides for every cardinal $\lambda$ a space $X_\lambda$ which is $\lambda$-cwH but not $\lambda^+$--cwH. So if we hope to get any level of compactness for the the property of being cwH, we have to restrict the class of spaces we consider. A fruitful case is the case where we restrict the local cardinality of the space. A motivating result is the construction by Shelah (using supercompact cardinal) of a model of Set Theory in which a space which is locally countable and which is $\omega_2$--cwH is cwH.

Can the Shelah result be generalized to larger cardinals , e.g. can you get a model in which for spaces which are locally of cardinality $\leq \omega_1$ and which are $\omega_3$-cwH are cwH? In general for which pair of cardinals $(\lambda, \mu)$ we can have models in which a space which is locally of cardinality $< \mu$ and which is $\lambda$--cwH are $\lambda^+$--cwH? In this lecture we shall give few examples where we get some ZFC theorems showing that for some pairs $(\lambda, \mu)$ compactness necessarily fails, and cases of pairs for which one can consistently have compactness for the property of being cwH.

Jordi Lopez Abad
Ramsey properties of embeddings between finite dimensional normed spaces

Given d=m , let E m,n be the set of all m×d matrices (a i,j ) such that
(a) ? d j=1 |a i,j |=1 for every 1=i=m .
(b) max m i=1 |a i,j |=1 for every 1=j=d .

These matrices correspond to the linear isometric embeddings from the normed space l d 8 :=(R d ,?·? 8 ) into l d 8 , in their unit bases.
We will discuss and give (hints of) a proof of the following new approximate Ramsey result:
For every integers d , m and r and every e>0 there exists n such that for every coloring of E d,n into r -many colors there is A?E m,n and a color i<r such that A·E d,m ?(c -1 (i)) e . Its proof uses the Graham-Rothschild Theorem on partitions of finite sets. We extend this result, first for embeddings between \emph{polyhedral} normed spaces, and finally for arbitrary finite dimensional normed spaces to get the following:
For every finite dimensional normed spaces E and F , every ?>1 and e>0 , and every integer r , there is some n such that for every coloring of Emb ? 2 (F,l n 8 ) into r -many colors there is T?Emb ? (G,l n 8 ) and some color i<r such that T°Emb ? (F,G)?(c -1 (i)) ? 2 -1+e .
As a consequence, we obtain that the group of linear isometries of the Gurarij space is extremely amenable. A similar result for positive isometric embeddings gives that the universal minimal flow of the group of affine homeomorphisms of the Poulsen simplex is the Poulsen simplex itself.
This a joint work (in progress) with Dana Bartosova (University of Sao Paulo) and Brice Mbombo (University of Sao Paulo)

Speaker 1 (from 12:30 to 13:30):
Dana Bartosova
Finite Gowers' Theorem and the Lelek fan

The Lelek fan is a unique non-degenrate subcontinuum of the Cantor fan with a dense set of endpoints. We denote by $G$ the group of homeomorphisms of the Lelek fan with the compact-open topology. Studying the dynamics of $G$, we generalize finite Gowers' Theorem to a variety of operations and show how it applies to our original problem. This is joint work with Aleksandra Kwiatkowska.

Speaker 2 (from 13:30 to 15:00):
Assaf Rinot
Productivity of higher chain condition

We shall survey the history of the study of the productivity of the k-cc in partial orders, topological spaces, and Boolean algebras. We shall address a conjecture that tries to characterize such a productivity in Ramsey-type language. For this, a new oscillation function for successor cardinals, and a new characteristic function for walks on ordinals will be proposed and investigated.

Sheila Miller
Critical sequences of rank-to-rank embeddings and a tower of finite left distributive algebras

In the early 1990's Richard Laver discovered a deep and striking correspondence between critical sequences of rank-to-rank embeddings and finite left distributive algebras on integers. Each $A_n$ in the tower of finite algebras can be defined purely algebraically, with no reference to the elementary embeddings, and yet there are facts about the Laver tables that have only been proven from a large cardinal assumption. We present here some of Laver's foundational work on the algebra of critical sequences of rank-to-rank embeddings and some work of the author's, describe how the finite algebras arise from the large cardinal embeddings, and mention several related open problems.

Ilijas Farah
Omitting types in logic of metric structures is hard

One of the important tools for building models with prescribed second-order properties is the omitting types theorem. In logic of metric structures omitting types is much harder than in classical first-order logic (it is Pi-1-1 hard). Although the motivation for this work comes from C*-algebras, the talk will mostly be on descriptive set theory. The intended takeaway from the talk is "logic of metric structures blends with descriptive set theory beautifully." This is joint work with Menachem Magidor.

No seminar

Daniel Soukup
Trees, ladders and graphs

The chromatic number of a graph $G$ is the least (cardinal) number $\kappa$ such that the vertices of $G$ can be covered by $\kappa$ many independent sets. A fundamental problem of graph theory asks how large chromatic number affects structural properties of a graph and in particular, is it true that a graph with large chromatic number has certain obligatory subgraphs? The aim of this talk is to introduce a new and rather flexible method to construct uncountably chromatic graphs from non special trees and ladder systems. Answering a question of P. Erdos and A. Hajnal, we construct graphs of chromatic number $\omega_1$ without uncountable infinitely connected subgraphs.

Konstantinos Tyros
A disjoint union theorem for trees

In this talk we will present an infinitary disjoint union theorem for level products of trees. An easy consequence of the dual Ramsey theorem due to T.J. Carlson and S.G. Simpson is that for every Suslin measurable finite coloring of the power set of the natural numbers, there exists a sequence $(X_n)_{n\in\mathbb{N}}$ of disjoint non-empty subsets of $\mathbb{N}$ such that the set
\[\Big\{\bigcup_{n\in Y}X_n:\; Y\;\text{non-empty subset of }\mathbb{N}\Big\}\]
is monochromatic. The result that we will present is of this sort, where the underline structure is the level product of a finite sequence of uniquely rooted and finitely branching trees with no maximal nodes of height $\omega$ instead of the natural numbers.
As it is required by the proof of the above result, we develop an analogue of the infinite dimensional version of the Hales--Jewett Theorem for maps defined on a level product of trees, which we will also present, if time permits.

Diana Ojeda
Finite forms of Gowers' Theorem on the oscillation stability of $c_0$

We give a constructive proof of the finite version of Gowers' $FIN_k$ Theorem and analyze the corresponding upper bounds. The $FIN_k$ Theorem is closely related to the oscillation stability of $c_0$. The stabilization of Lipschitz functions on arbitrary finite dimensional Banach spaces was proved well before by V. Milman. We compare the finite $FIN_k$ Theorem with the Finite Stabilization Principle found by Milman in the case of spaces of the form $\ell_{\infty}^n$, $n\in N$, and establish a much slower growing upper bound for the finite stabilization principle in this particular case.

Seminar cancelled

Mike Pawliuk
Various types of products of Fraisse Classes, various types of amenability and various types of preservation results.
This is joint work with Miodrag Sokic.

In a recent paper of Jakub Jasinski, Claude Laflamme, Lionel Nguyen Van Thé and Robert Woodrow (Arxiv: 1310.6466) it was shown that certain Fraisse Classes are actually Ramsey classes. For many of those cases we have determined whether their automorphism groups are extremely amenable or not. Some of these spaces turn out to actually be a special type of product of Fraisse classes. We were able to prove that unique ergodicity (a type of amenability) is preserved under this type of product.

Martino Lupini
Functorial complexity of Polish and analytic groupoids

I will explain how one can generalize the theory of Borel complexity from analytic equivalence relations to groupoids by means of the notion of Borel classifying functor. This framework allows one to capture the complexity of classifying the objects of a category in a functorial way. I will then present the first results relating the functorial complexity of a groupoid and the complexity of its associated orbit equivalence relation, focusing on the case of Polish groupoids: For Polish groupoids with essentially treeable equivalence relations any Borel reduction between the orbit equivalence relations extends to a Borel classifying functor. On the other hand for any countable non-treeable equivalence relation E there are Polish groupoids of different functorial complexity both having E as associated orbit equivalence relation. The proof of these results involves a generalization of some fundamental results on the descriptive set theory of actions of Polish groups --such as the Becker-Kechris theorem on Polishability of Borel G-spaces-- to actions of Polish groupoids.

Saeed Ghasemi
An analogue of Feferman-Vaught theorem for reduced products of metric structures

I will give a metric version of the Feferman-Vaught theorem for reduced products of discrete spaces. We will use this to show that, under the continuum hypothesis, the reduced powers of any metric structure over atomless layered ideals are isomorphic. As another application, I will give an example of two reduced products of sequences of matrix algebras over Fin, which are elementarily equivalent, therefore isomorphic under the CH, with no trivial isomorphisms between them.

Christopher Eagle
Model theory of abelian real rank zero C*-algebras

We consider algebras of the form $C(X)$, where $X$ is a $0$-dimensional compact Hausdorff space, from the point of view of continuous model theory. We characterize these algebras up to elementary equivalence in terms of invariants of the Boolean algebra $CL(X)$ of clopen subsets of $X$. We also describe several saturation properties that $C(X)$ may have, and relate these to topological properties of $X$ and saturation of $CL(X)$. We will discuss some consequences of saturation when we view $C(X)$ as a $C^*$-algebra. All the necessary background on continuous logic will be provided. This is joint work with Alessandro Vignati.