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1.
Drees, Kevin Michael.
<i>C_{p}</i>(<i>X</i>,Z).

Degree: PhD, Mathematics, 2009, Bowling Green State University

URL: http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1243803693

We examine the ring of continuous
integer-valued continuous functions on a topological space X,
denoted <i>C</i>(<i>X</i>,Z), endowed with
the topology of pointwise convergence, denoted
<i>C_{p}</i>(<i>X</i>,Z).
We first deal with the basic properties of the ring
<i>C</i>(<i>X</i>,Z) and the space
<i>C_{p}</i>(<i>X</i>,Z).
We find that the concept of a zero-dimensional space plays an
important role in our studies. In fact, we find that one need only
assume that the domain space is zero-dimensional; this is similar
to assume the space to be Tychonoff when studying
<i>C</i>(<i>X</i>), where
<i>C</i>(<i>X</i>) is the ring of
real-valued continuous functions. We also find the space
<i>C_{p}</i>(<i>X</i>,Z)
is itself a zero-dimensional space. Next, we
consider some specific topological properties of the space
<i>C_{p}</i>(<i>X</i>,Z)
that can be characterized by the topological properties of X. We
show that if
<i>C_{p}</i>(<i>X</i>,Z)
is topologically isomorphic to
<i>C_{p}</i>(<i>Y</i>,Z),
then the spaces <i>X</i> and <i>Y</i> are
homeomorphic to each other, this is much like a the theorem by
Nagata from 1949. We show that if <i>X</i> is a
zero-dimensional space, then there is a zero-dimensional space
<i>Y</i> such that <i>X</i> is embedded in
<i>C_{p}</i>(<i>Y</i>,Z).
Thus every zero-dimensional space can be viewed as a collection of
integer-valued continuous functions. We consider and prove the
collection of all linear combinations of characteristic functions
on clopen (open and closed) subsets is a dense subspace of
<i>C_{p}</i>(<i>X</i>,Z).
We then consider when the space
<i>C_{p}</i>(<i>X</i>,Z)
are <i>G</i>_{d}- and
<i>F</i>_{¿}-subsets of the
collection of all functions from <i>X</i> to Z (a
<i>G</i>_{d}-subset is a countable
intersection of open subsets and a
<i>F</i>_{¿}-subset is a countable
union of closed subsets). We make classifications
for when
<i>C_{p}</i>(<i>X</i>,Z)
is a discrete space, metrizable space, Frechet-Urysohn space,
sequential space, and <i>k</i>-space. We end with some
results on cardinal invariants and the relationships between the
tightness and Lindelöf numbers of related
spaces.
*Advisors/Committee Members: McGovern, Warren Wm. (Advisor).*

Subjects/Keywords: Mathematics; pointwise topology; rings of continuous functions; zero-dimensional; weight; character; metrizable space; Frechet-Urysohn space

…point
in time. It is know when Cp (X) is metrizable, a Fr´
echet-*Urysohn* *space*, a… …metrizable, is a
4
Fr´
echet-*Urysohn* *space*, when it is a sequential and when it is a k-*space*… …2
that the *space* X possesses. When studying C(X) it is usually assumed that the… …*space* X
is a Tychonoff *space*, since given any *space* X there exists a Tychonoff *space* Y such… …that
C(X) is ring isomorphic to C(Y ). Recall that a *space* X is Tychonoff…

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Drees, K. M. (2009). <i>C_{p}</i>(<i>X</i>,Z). (Doctoral Dissertation). Bowling Green State University. Retrieved from http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1243803693

Chicago Manual of Style (16^{th} Edition):

Drees, Kevin Michael. “<i>C_{p}</i>(<i>X</i>,Z).” 2009. Doctoral Dissertation, Bowling Green State University. Accessed April 02, 2020.
http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1243803693.

MLA Handbook (7^{th} Edition):

Drees, Kevin Michael. “<i>C_{p}</i>(<i>X</i>,Z).” 2009. Web. 02 Apr 2020.

Vancouver:

Drees KM. <i>C_{p}</i>(<i>X</i>,Z). [Internet] [Doctoral dissertation]. Bowling Green State University; 2009. [cited 2020 Apr 02].
Available from: http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1243803693.

Council of Science Editors:

Drees KM. <i>C_{p}</i>(<i>X</i>,Z). [Doctoral Dissertation]. Bowling Green State University; 2009. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1243803693

2. Conant, Gabriel J. Model Theory and Combinatorics of Homogeneous Metric Spaces.

Degree: 2015, University of Illinois – Chicago

URL: http://hdl.handle.net/10027/19725

We develop the model theory of generalized metric spaces, in which distances between points are taken from arbitrary ordered additive structures. Our focus is on the class of structures obtained by fixing a countable, positively ordered monoid R, and considering the universal, ultrahomogeneous Urysohn space U(R), which takes distances in R. As notable examples, this class includes the Fraisse limits of rational metric spaces, graphs, and refining equivalence relations, as well as more general structures previously studied by Delhomme, Laflamme, Pouzet, and Sauer. We characterize quantifier elimination in the theory of U(R) (in a relational language) using continuity properties in R, which hold in most natural examples arising in previous literature. In the case where quantifier elimination holds, we consider classification theoretic “neostability” properties of U(R), focusing especially on when such properties are characterized by first-order statements about the monoid R. As a first example of this phenomenon, we prove that U(R) is stable if and only if it is an ultrametric space. Using a characterization of forking for complete types, we are also able to characterize simplicity in terms of “low complexity” in the monoid R. We then generalize the characterizations of stability and simplicity, in order to define the integer-valued “archimedean complexity” of an ordered monoid. We show that the position of U(R) in Shelah's hierarchy of strong order properties is precisely determined by this invariant, which gives the first class of natural examples in which the entirety of this hierarchy can be meaningfully interpreted. We also generalize previous work of Casanovas and Wagner to obtain necessary conditions for elimination of hyperimaginaries and weak elimination of imaginaries in U(R). The last two chapters shift in focus to the combinatorics of these Urysohn spaces. We generalize work of Solecki to prove that, when R is archimedean, the class of finite R-metric spaces has the Hrushovski property. We then extend this result to a larger class, which includes ultrametric spaces. Finally, we consider the asymptotic behavior of finite distance monoids, and uncover some surprising connections to other areas of algebraic and additive combinatorics.
*Advisors/Committee Members: Marker, David (advisor), Baldwin, John (committee member), Goldbring, Isaac (committee member), Rosendal, Christian (committee member), Malliaris, Maryanthe (committee member).*

Subjects/Keywords: model theory; classification theory; generalized metric space; Urysohn space; stability; simplicity; strong order property; extending isometries

…graph, and the rational *Urysohn* *space*. These structures arise as motivational examples in many… …the rational *Urysohn*
*space*), and we will show that, moreover, this class exhibits the… …particular, we consider generalizations of the rational
*Urysohn* *space* obtained by constructing a… …class of R-*Urysohn* spaces, which includes the case when UR is
an ultrametric *space*. This… …independence property
RUS
R-*Urysohn* spaces, where R is a *Urysohn* monoid
SOP
strict order property…

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Conant, G. J. (2015). Model Theory and Combinatorics of Homogeneous Metric Spaces. (Thesis). University of Illinois – Chicago. Retrieved from http://hdl.handle.net/10027/19725

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Conant, Gabriel J. “Model Theory and Combinatorics of Homogeneous Metric Spaces.” 2015. Thesis, University of Illinois – Chicago. Accessed April 02, 2020. http://hdl.handle.net/10027/19725.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Conant, Gabriel J. “Model Theory and Combinatorics of Homogeneous Metric Spaces.” 2015. Web. 02 Apr 2020.

Vancouver:

Conant GJ. Model Theory and Combinatorics of Homogeneous Metric Spaces. [Internet] [Thesis]. University of Illinois – Chicago; 2015. [cited 2020 Apr 02]. Available from: http://hdl.handle.net/10027/19725.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Conant GJ. Model Theory and Combinatorics of Homogeneous Metric Spaces. [Thesis]. University of Illinois – Chicago; 2015. Available from: http://hdl.handle.net/10027/19725

Not specified: Masters Thesis or Doctoral Dissertation

3. Slutskyy, Kostyantyn. Extending partial isomorphisms.

Degree: PhD, 0439, 2012, University of Illinois – Urbana-Champaign

URL: http://hdl.handle.net/2142/30971

There are two main topics in the thesis. In the second chapter we study two-dimensional classes of topological similarity in the groups of automorphisms of some linearly ordered Fraisse classes: the rationals, the linearly ordered random graph and the linearly ordered Urysohn space. The main theorem establishes meagerness of two-dimensional similarity classes in these groups. As a byproduct we get some results about the group of isometries of the Urysohn space.
The third chapter is devoted to the metrics on the free products and HNN extensions of groups with two-sided invariant metrics. Using the approach of Graev to metrics on the free groups we show the existence of the coproducts in the category of groups with two-sided invariant metrics and Lipschitz homomorphisms. We then apply this theory to formulate a criterion when two topologically similar elements in a SIN Polish group are conjugate inside a bigger SIN Polish group.
*Advisors/Committee Members: Rosendal, Christian (advisor), van den Dries, Lou (Committee Chair), Solecki, Slawomir (committee member), Rosendal, Christian (committee member), Kapovitch, Ilia (committee member).*

Subjects/Keywords: Polish groups; Graev metrics; topological similarity; induced conjugacy classes; Urysohn space

…rationals with the usual ordering, and the rational *Urysohn* metric *space*.
Since the work of Fra… …ordered Rado graph, and the ordered rational *Urysohn* *space*.
The main tool in the proof of the… …results about the structure of the group
of isometries of the rational *Urysohn* *space*. For a… …*Urysohn*
*space*, and let U be the *Urysohn* *space*, which is the metric completion of QU.
J. Melleray… …interest for us.
The *Urysohn* *space* U is a complete separable metric *space*, that is uniquely…

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Slutskyy, K. (2012). Extending partial isomorphisms. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/30971

Chicago Manual of Style (16^{th} Edition):

Slutskyy, Kostyantyn. “Extending partial isomorphisms.” 2012. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed April 02, 2020. http://hdl.handle.net/2142/30971.

MLA Handbook (7^{th} Edition):

Slutskyy, Kostyantyn. “Extending partial isomorphisms.” 2012. Web. 02 Apr 2020.

Vancouver:

Slutskyy K. Extending partial isomorphisms. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2012. [cited 2020 Apr 02]. Available from: http://hdl.handle.net/2142/30971.

Council of Science Editors:

Slutskyy K. Extending partial isomorphisms. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2012. Available from: http://hdl.handle.net/2142/30971