Title

Cp(X,ℤ)

Author

Kevin Drees, Bowling Green State University

Date of Award

2009

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics

First Advisor

Warren Wm. McGovern, PhD

Second Advisor

Christopher Rump, PhD (Committee Member)

Third Advisor

Rieuwert J. Blok, PhD (Committee Member)

Fourth Advisor

Kit C. Chan, PhD (Committee Member)

Abstract

We examine the ring of continuous integer-valued continuous functions on a topological space X, denoted C(X,ℤ), endowed with the topology of pointwise convergence, denoted C_p(X,ℤ). We first deal with the basic properties of the ring C(X,ℤ) and the space C_p(X,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 C(X), where C(X) is the ring of real-valued continuous functions. We also find the space C_p(X,ℤ) is itself a zero-dimensional space.

Next, we consider some specific topological properties of the space C_p(X,ℤ) that can be characterized by the topological properties of X. We show that if C_p(X,ℤ) is topologically isomorphic to C_p(Y,ℤ), then the spaces X and Y are homeomorphic to each other, this is much like a the theorem by Nagata from 1949. We show that if X is a zero-dimensional space, then there is a zero-dimensional space Y such that X is embedded in C_p(Y,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 C_p(X,ℤ). We then consider when the space C_p(X,ℤ) are G_δ- and F_ς-subsets of the collection of all functions from X to ℤ (a G_δ-subset is a countable intersection of open subsets and a F_ς-subset is a countable union of closed subsets).

We make classifications for when C_p(X,ℤ) is a discrete space, metrizable space, Frechet-Urysohn space, sequential space, and k-space. We end with some results on cardinal invariants and the relationships between the tightness and Lindelöf numbers of related spaces.

Recommended Citation

Drees, Kevin, "Cp(X,ℤ)" (2009). Mathematics Ph.D. Dissertations. 4.
https://scholarworks.bgsu.edu/math_diss/4