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 Cp(X,ℤ). We first deal with the basic properties of the ring C(X,ℤ) and the space Cp(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 Cp(X,ℤ) is itself a zero-dimensional space.
Next, we consider some specific topological properties of the space Cp(X,ℤ) that can be characterized by the topological properties of X. We show that if Cp(X,ℤ) is topologically isomorphic to Cp(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 Cp(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 Cp(X,ℤ). We then consider when the space Cp(X,ℤ) are Gδ- and Fς-subsets of the collection of all functions from X to ℤ (a Gδ-subset is a countable intersection of open subsets and a Fς-subset is a countable union of closed subsets).
We make classifications for when Cp(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.