This book illustrates the concept of neighbourhood in the beginning and open sets and interior points are defined via neighbourhood. The proofs of results relating to open sets and interior points are modified accordingly. Closed sets are defined as the complement of an open set and the necessary changes in the statements of the results and their proofs on closed sets are made. Key features include: the concept of equivalent metrics; motivation for homeomorphism; a number of results, examples and counter examples along with examples of non-metric spaces; numerous problems and their solutions; and geometrical interpretation of open spheres in R and their figures are added.