## Date of Award

7-1964

## Degree Type

Master's Essay - Restricted

## Degree Name

Master of Science (MS)

## Department

Mathematical and Statistical Sciences

## Abstract

This paper was prompted by a still unsolved problem posed in 1920 by a Russian mathematician, M. Souslin.

The problem was: If a set X is linearly ordered, has no first or last element, is separable and if every collection of disjoint open intervals in X is countable, does there exist a one to one, onto, order preserving mapping from the set X to the real line.

A set is to be considered as a collection of objects. The elements of most of the sets considered here will be points. The particular set which contains no objects is to be referred to as the empty set or the null set. The empty set is a subset of every set. The complement of a set A is the set A' where A' is composed of all the elements not in A but which are in the consideration.

## Recommended Citation

Loop, Robert A., "Some Properties of Metric and Hausdorff Spaces" (1964). *Master's Essays (1922 - )*. 1633.

https://epublications.marquette.edu/essays/1633