It is indeed appropriate to ask "why open sets in topology ?" However, the answer is not so simple precisely since the concept of topology proves to be far more deep and complex than Hausdorff, Kuratowski or Bourbaki have ever imagined, and which may call the HKB topology. For starters, it turned out decades ago that the usual, open set based, that is, HKB topology leads to a category which is not Cartesian closed. And this creates serious difficulties when dealing with topologies on function spaces, and in particular, in duality theory for locally convex spaces. More simply, and without categories, there are most important topological type processes in mathematics which simply cannot be described by HKB topologies. Such are, for instance, in measure theory and partially ordered spaces. As a consequence, various more general concepts of pseudo-topologies have been suggested. What happened, however, is that the doors thus opened up proved to be too large for those who tried to pursue them, or would have liked to use them ... In other words, topologies beyond HKB are a far less cozy venture than usually customary in mathematics ... Some details about the above and relevant references may be found in

arXiv:1001.1866 [pdf, ps, other] Title: Beyond Topologies, Part I Authors: Elemer E Rosinger, Jan Harm van der Walt Subjects: General Mathematics (math.GM)

arXiv:1005.1243 [pdf, ps, other] Title: Rigid and Non-Rigid Mathematical Theories: the Ring $\mathbb{Z}$ Is Nearly Rigid Authors: Elemer E. Rosinger Subjects: General Mathematics (math.GM)

It is indeed appropriate to ask "why open sets in topology ?" However, the answer is not so simple precisely since the concept of topology proves to be far more deep and complex than Hausdorff, Kuratowski or Bourbaki have ever imagined, and which may call the HKB topology. For starters, it turned out decades ago that the usual, open set based, that is, HKB topology leads to a category which is not Cartesian closed. And this creates serious difficulties when dealing with topologies on function spaces, and in particular, in duality theory for locally convex spaces. More simply, and without categories, there are most important topological type processes in mathematics which simply cannot be described by HKB topologies. Such are, for instance, in measure theory and partially ordered spaces. As a consequence, various more general concepts of pseudo-topologies have been suggested. What happened, however, is that the doors thus opened up proved to be too large for those who tried to pursue them, or would have liked to use them ... In other words, topologies beyond HKB are a far less cozy venture than usually customary in mathematics ... Some details about the above and relevant references may be found in

arXiv:1001.1866 [pdf, ps, other] Title: Beyond Topologies, Part I Authors: Elemer E Rosinger, Jan Harm van der Walt Subjects: General Mathematics (math.GM)

arXiv:1005.1243 [pdf, ps, other] Title: Rigid and Non-Rigid Mathematical Theories: the Ring $\mathbb{Z}$ Is Nearly Rigid Authors: Elemer E. Rosinger Subjects: General Mathematics (math.GM)