Chapter 32 (Connectedness)

olivierd

Jul 22, 2017 2 min read

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…

Category: Chapter 32 (Connectedness)

Connected subsets one of which meets the closure of the other

Union of transitively overlapping connected spaces