Homeomorphic spaces in the plane

The idea of topological equivalence, or homeomorphism, is one of the most basic in any study of topology. In this paper, pairs of spaces imbedded in the plane are compared to determine whether or not they are homeomorphic. The decisions are based on four main properties. In Chapter I, homeomorphism is defined and the conditions under which a property is said to be a topological property are given. Chapter II deals with compactness, the first of the four properties previously mentioned. Examples of compact and non-compact spaces are given and then used to investigate the topological equivalence of some well-known spaces. The second of the four properties, connectedness, is defined and discussed in Chapter III. It is shown that connectedness is a topological property, and examples of homeomorphic and non-homeomorphic spaces are given. The third of the four properties, that of having a point of separation, is presented in Chapter IV. It is proved that this is indeed a topological property, and examples of sets with this property are studied