Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Survey

Document related concepts

Transcript

Introduction to Topology University of Memphis Fall 2008 Dwiggins Homework Assignment # 5 Due Thursday 30 October # 1. Exercises from textbook, page 67, # 5, # 6, # 7, as discussed in class. (# 6 is the bonus, showing x2 is not a contraction under dX.) # 2. Exercises from textbook, page 68, # 14, # 15. # 3. Let (X, be a topological space. Prove X is T1 if and only if every singleton {x0} is closed under , for every x0 X. Next use this to prove every finite T1 space must have the discrete topology. (This is exercise # 5 on page 84). Why can’t the same proof be used to prove every T1 space must be discrete? # 4. Prove T1 is hereditary, and also prove regularity is hereditary, so that T3 (= regular + T1) is also hereditary. # 5. Let X = R and = {(a, b] : a < b}, and let be the topology generated by . That is, is a subbase for , formed by taking intersections of finitely many elements from , and is a base for , with open sets formed by taking arbitrary unions of elements of . (a) Prove = {}, so that every half-open interval (a, b] is a basic open set in this topology. [Hint: Given S1, S2 show that if B = S1 S2 then either B = or B = S3 (b) Use properties B1 and B2 (see Theorem 4.20) to prove is a base. (c) Prove the generated topology is Hausdorff, i.e. given x R, y R, x ≠ y, S1 S2 with x S1y S2S1 S2 = . (In terms of x and y, calculate the values of a and b needed for each S in order for this to be true.) # 6. Let (X, be a Hausdorff space. (a) Given three distinct points x, y, z X, show there exist three open sets U, V, W with x U, y V, z W, U V = , U W = , V W = . (b) Show how part (a) extends to the following: Given n N and n distinct points {xk : 1 < k < n} X, there exist sets {Gk : 1 < k < n} open in X with xk Gk for k = 1, . . ., n and Gi Gj = for i ≠ j. (c) Now suppose (X, is an infinite Hausdorff space. Prove infinitely many open sets {Gk : k N} such that Gi Gj = for i ≠ j. # 7. Prove normality is a topological property, i.e. if X and Y are homeomorphic then X is normal if and only if Y is normal. In the proof, assume the existence of a homeomorphism f : X Y. Give an example showing normality need not be conserved if f is merely continuous.