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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  S1y  S2S1  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.