Studies on C Spaces

Abstract

The concept of connectedness is discussed in numerous mathematical fields such as topology and graph theory and it is highly applicable in image filter-ing and segmentation, image compression and coding, motion analysis, pattern recognition etc. Both a topological and a graph theoretical framework are used to characterize connectivity. However, there are sometimes differences between topology and graph theory approaches to connectedness. But complications arise when these two approaches are used independently. In both theory and practice, a general description of connectedness that is applicable to both graph theory and topology is more beneficial. In 1983, R. Börger introduced an axiomatic approach to connectivity in order to standardize the definition of connectedness across these mathematical domains. The axioms were certain characteristics of connected sets such as empty set and singletons are connected and that union of connected sets having nonempty intersection is connected. A collection of sub- sets of a set satisfying these two axioms is called a c-structure and a set together with a c-structure on it is called a c-space. The objective of this thesis is to present new contributions to the theory of c- spaces. Our primary focus is on the study of order-induced c-space, which is the c-space obtained from a linearly ordered set. Here, we discuss topological order induced c-spaces. We also characterize complete linearly ordered sets and dense linearly ordered sets in relation to order-induced c-space. Then we investigate the reversible property of c-spaces. The reversible c-spaces are characterized and prove the existence of non-reversible c-spaces with any infinite cardinality. Further, we define cut-point c-spaces and investigate the features of cut-point c-spaces. Moreover, we construct a cut-point c-structure on the union of an arbitrary family of mutually disjoint c-spaces if at least one of these c-spaces is a cut-point c-space. Finally, we associate c-spaces with hypergraphs. We discuss the properties of c-structures obtained from hypergraphs. Also, we prove that its members are the vertex sets of the connected hypersubgraphs of the given hypergraph.