## Mathematics Ph.D. Dissertations

# Cp(X,ℤ)

## Date of Award

2009

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy (Ph.D.)

## Department

Mathematics

## First Advisor

Warren Wm. McGovern

## Second Advisor

Christopher Rump (Committee Member)

## Third Advisor

Rieuwert J. Blok (Committee Member)

## Fourth Advisor

Kit C. Chan (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