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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.