Topological Constructions

Abstract

The main object of this chapter is the construction of “quotient spaces”. The motivation for this comes from the construction of models of some geometrical objects such as circle, cylinder, Möbius band, etc., by gluing things together. In Sect. 6.1, we translate the process of gluing into precise mathematical language by using the notion of equivalence relation and study the “quotient topology” for the set of equivalence classes of a topological space with an equivalence relation. Of course, quotient spaces could have been discussed just after studying subspaces and products; however, comprehending the discussion becomes much easier after having seen some results about compactness. Moreover, it is often more convenient to describe quotient spaces by means of “identification maps”. This approach will be adopted in Sect. 6.2. In the next three sections, we develop several interesting techniques for producing new spaces from old ones by combining different constructions. These methods of generating new spaces are of great importance in the study of algebraic topology. In the last section, we will study topology induced by functions from a given set into a collection of topological spaces and its dual notion. In particular, we will see the concept of topology generated by subspaces of a given space.