All vector spaces are by default over or . We consider three vector spaces , and . Let be a tensor product of and , and a tensor product of and . The aim is first to show that , together with the mapping: is a tensor product of , and . Proof:…
Exercise 225 (Chapter 32, “Connectedness”): Prove the following generalization of theorem 40. Let ( in ) be a collection of connected subsets of topological space X, and let the equivalence relation on this collection of sets generated by if have just one equivalence class. Then is connected. For clarity, rather than a family I will…
Exercise 7 (Chapter 2, “Categories”): In the category of sets, the two morphisms ( and ) in a direct product are monomorphisms and the two morphisms in a direct sum are epimorphisms. Is this true in every category? There are two mistakes in this wording. The author has exchanged “monomorphisms” and “epimorphisms”. In a direct…
Exercise 185 (Chapter 28, “The category of topological spaces”): Prove that both the direct product and the direct sum of two Hausdorff topological spaces is Hausdorff. We will consider Hausdorff topological spaces and and their direct sum (in the first part) or product (in the second part) . We will proceed without reference to the…
Exercise 182 (Chapter 29, “The category of topological spaces”): Give a universal definition which leads to the introduction of the indiscrete topology on a set. On a set , the indiscrete topology is the unique topology such that for any set , any mapping is continuous. This isn’t really a universal definition. We have seen…
Exercise 105 (Chapter 17, “Functors”): Define the identity functor from a category to that same category. What do you suppose is meant by equivalent categories? Identity functors We have encountered two flavors of identity definitions. Identity mappings on sets are defined by the way they act on set elements, namely that they don’t change them….