... et is again open (closed).Note: In a topological space, (iii) is used as definition for continuity.However, in general (ii) and (iii) will no longer be equivalent unless one usesgeneralized ... unions. In general, a space X together with a family of sets O, the open sets,satisfying (i) –(iii) is called a topological space. The notions of interiorpoint, limit point, and neighborhood ... in infinite dimensional spaces different norms will giveraise to different convergent sequences! In fact, the key to solving prob-lems in infinite dimensional spaces is often finding the right norm!...