## Connected subsets one of which meets the closure of the other

Exercise 224 (Chapter 32, “Connectedness”): Let and be connected subsets of topological space such that intersects . Prove that is connected. Find an example to show that it is not enough to assume that intersects . Let us assume that and are open subsets of such that is empty and that . To show that…

## Union of transitively overlapping connected spaces

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…

## Direct products/sums: when the associated morphisms are epi/mono

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…

## Direct sum of Hausdorff topological spaces

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…