On the Construction of special metrics involving Levenshtein and Hamming distances

Metric spaces are one of the useful types of topological spaces in measurement of length, and distance between points in underlying sets. They are useful for much more abstract and irregular sets than intervals in R 2 or R 3 . The ability to measure and compare distances between elements is often crucial and metric spaces provide more structure than general topological spaces. We begin by giving a brief introduction and background to topological spaces in general. We then go ahead to discuss metric spaces with a quick overview on concepts like connectedness, compactness, continuity and completeness and define formally what a metric is and give the basic properties of metric spaces. We state without proof some theorems in topology which we invoke subsequently in the construction of some metric spaces. A large class of metric spaces exists and this work illustrates the construction of some of these metric including the Hamming distance, the Levenshtein distance, the Hausdorff metric and other applications of metric spaces.