The purpose of this paper will be to investigate certain properties of algebraic systems known as rings.

The purpose of this paper is to investigate some properties of compact topological spaces and to relate these concepts to the separation properties.

The purpose of this composition is to develop a rigorous, axiomatic proof of the comparability of the cardinals of infinite sets.

The purpose of this paper is to examine properties of convex sets in linear topological spaces with special emphasis on compact convex sets.

The purpose of this paper is to investigate Saunders MacLane's axioms for plane geometry. The wording of the axioms has been modified; however, the concept suggested by each axiom remains the same.

In the category of mathematics called partial differential equations there is a particular type of problem called the Dirichlet problem. Proof is given in many partial differential equation books that every Dirichlet problem has one and only one solution. The explicit solution is very often not easily determined, so that a method for approximating the solution at certain points becomes desirable. The purpose of this paper is to present and investigate one such method.

This thesis discusses the convergence of nets through a series of theorems and proofs.

This paper is devoted to finding integral solutions of algebraic equations. Only algebraic equations with integral coefficients are considered. The elementary properties of integers are assumed.

The main purpose of this paper is the study of transformations in topological space and relationships between special types of transformations.

The study of metric spaces is closely related to the study of topology in that the study of metric spaces concerns itself, also, with sets of points and with a limit point concept based on a function which gives a "distance" between two points. In some topological spaces it is possible to define a distance function between points in such a way that a limit point of a set in the topological sense is also a limit point of the same set in a metric sense. In such a case the topological space is "metrizable". The real numbers with its usual topology is an example of a topological space which is metrizable, the distance function being the absolute value of the difference of two real numbers. Chapters II and III of this thesis attempt to classify, to a certain extent, what type of topological space is metrizable. Chapters IV and V deal with several properties of metric spaces and certain functions of metric spaces, respectively.