Download Warm-up questions Worksheet problems Assignment questions and more Exams Calculus in PDF only on Docsity! Math 490 Homework #12 Friday 3 December 2021 Warm-up questions (These warm-up questions are optional, and wonโt be graded.) 1. Let (X, T ) be a topological space. Show that any subset A = {x} โ X of a single element is connected. 2. Let X = {a, b, c, d} with the topology T = {โ
, {a}, {a, b}, {c}, {a, c}, {a, b, c}, {a, b, d}, {a, b, c, d}}. Is X connected? 3. (a) Show that, for a, b โ R, the subsets โ
, {a}, (a, b), (a, b], [a, b), [a, b], (a,โ), [a,โ), (โ, b), (โ, b], and R of R are all intervals in the sense of Problem b. (b) Show that every interval must have one of these forms. 4. Give an example of a subset A of R (with the standard topology) such that A is not connected, but A is connected. (Compare to Assignment Problem 1) Worksheet problems (Hand these questions in!) โข Worksheet #15 Problems 3, 5, 6(b). Assignment questions (Hand these questions in!) 1. Let (X, TX) be a topological space, and let A โ X be a connected subset. Let B be any subset such that A โ B โ A. Prove that B is connected. Remark: This shows in particular that if A is connected, then so is A. 2. (a) Let (X, TX) and (Y, TY ) be topological spaces. Let f : X โ Y be a continuous map. Prove that if X is connected, then f(X) is connected. In other words, the continuous image of a connected space is connected. (b) Recall the Intermediate Value Theorem from real analysis (which you may use without proof). Intermediate Value Theorem. If f : [a, b] โ R is continuous and d lies between f(a) and f(b) (i.e. either f(a) โค d โค f(b) or f(b) โค d โค f(a)), then there exists c โ [a, b] such that f(c) = d. Define a subset A โ R to be an interval if whenever x, y โ A and z lies between x and y, then z โ A. Prove that any interval of R is connected. Hint: Worksheet #15 Problem 4. (c) Prove that any subset of R that is not an interval is disconnected. These last two results together prove: Page 1
