Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Topology, MM8002/SF2721, Spring 2017. Exercise set 2 Exercise 1. Let f : X → Y be a map. Show that the following are equivalent: • f is continous. • f −1 (C) is closed for all closed sets C ⊆ Y. • f −1 (A◦ ) ⊆ f −1 (A)◦ for all A ⊆ Y. • f −1 (A) ⊆ f −1 (A) for all A ⊆ Y. • The restriction f |U : U → Y is continous for all open subsets U ⊆ X. Exercise 2. Let f : X → Y be a continous bijective map. Show that the following are equivalent: • f is open, i.e. images of open sets are open. • f is closed, i.e. images of closed sets are closed. • f is a homeomorphism. • The restriction f |U : U → f (U ) is a homeomorphism for all open subsets U ⊆ X. Exercise 3. Let X be a topological space with the following property • For every point p ∈ X there exists a continous function f : X → R such that f −1 (0) = {p}. Show that X is Hausdorff. Exercise 4. Let X be a second countable space. Show that X contains a countable dense subset. Exercise 5. Consider the set X := (−∞, 0) ∪ {0, 00 } ∪ (0, ∞) = R ∪ {00 }. • Show that {(a, b)| a < b ∈ R} ∪ {(a, 0) ∪ {00 } ∪ (0, b)| a < 0 < b ∈ R} forms a basis for a topology. • Show that the topological space given by the basis above is not Hausdorff. • Show that every point in the topological space defined above has an open neighborhood that is homeomorphic to an open set in R with it’s usual metric topologty. • Show that the topological space defined above is second countable. Remark. The above is an example of a topological space that satisfies all the axioms of a manifold besides being Hausdorff. Can you give an example of a topological space that satisfies all the axioms of a manifold besides being second countable? How about a second countable space that is Hausdorff, but has a point that doesn’t have an open neighborhood that is homeomorphic to an open subset of Rn ? 1