| A density continuous function is defined as a continuous function from a Tychonoff space X into the real numbers with the density topology. The collection of density continuous functions on X is denoted by C(X, Rd ), and it is a subset of the set C(X) of real-valued continuous functions on X. The set C(X) is a ring and a lattice under pointwise operations, and it is shown that C(X, Rd ) is a sublattice of C(X). However, unlike C(X), C( X, Rd ) is not always a ring. For example, C( Rd , Rd ) is not even a group. One major result of this dissertation is a characterization of when C(X, Rd ) is a group and a ring. It is shown that C( X, Rd ) is a ring precisely when each density continuous function is locally constant, and in this case X is defined to be a density P-space. Examples and properties of density P-spaces are given.; The idea of a density P-space is generalized to density almost P-spaces, density F-spaces, density F'-spaces, density U-spaces, and density semi- F-spaces. Characterizations and properties of these spaces are given. These conditions on C(X, Rd ) are also compared to the analagous conditions with respect to C(X).; Lattice-theoretic properties of C(X, Rd ) are investigated. Another major result of this dissertation is a characterization of conditional completeness and conditional sigma-completeness of the lattice C(X, Rd ). When X is zero-dimensional, it is shown that the property of conditional sigma-completeness of C( X, Rd ) is equivalent to X being a P-space and conditional completeness of C(X, Rd ) is equivalent to X being an extremely disconnected P-space. Also, the lattice of density zerosets Zd[ X] is considered. |
