Q1/a/ Let X be a topological space and A, B = P(X) show that whether the statement is true or not int(A UB) = int(A) U int(B) int(AUB): b/ If ẞ is a collection of open sets in X Show that ẞ is a base for a topology o X iff for each x Є X the collection ẞx = {BEẞ,x E B} is a nbhd base at x. Q2/a/Let A be a subspace of a topological space X Show that: 1. ẞ is a base for X then {BNA, BE ẞ} is base for A. 2. HCA is open in A iff H = An U, U open in X. b\Define the coarser topology and finer topology and give example with prove: 1.The coarser topology than any topology 2. The finer topology than any topology. Q3/a/ If there exist an operation & on P(X) satisfying 1. Cl(0) = 0 2. A Cl(A) for any A = P(X) 3. CI(CI(A)) = CI(A) for any A = P(X) 4. CI(AUB) = CI(A) U CI(B) for any A, B = P(X) Show that there exist a unique topology T on X such that (A) = Cl(A) for any A Є P(X)
Q1/a/ Let X be a topological space and A, B = P(X) show that whether the statement is true or not int(A UB) = int(A) U int(B) int(AUB): b/ If ẞ is a collection of open sets in X Show that ẞ is a base for a topology o X iff for each x Є X the collection ẞx = {BEẞ,x E B} is a nbhd base at x. Q2/a/Let A be a subspace of a topological space X Show that: 1. ẞ is a base for X then {BNA, BE ẞ} is base for A. 2. HCA is open in A iff H = An U, U open in X. b\Define the coarser topology and finer topology and give example with prove: 1.The coarser topology than any topology 2. The finer topology than any topology. Q3/a/ If there exist an operation & on P(X) satisfying 1. Cl(0) = 0 2. A Cl(A) for any A = P(X) 3. CI(CI(A)) = CI(A) for any A = P(X) 4. CI(AUB) = CI(A) U CI(B) for any A, B = P(X) Show that there exist a unique topology T on X such that (A) = Cl(A) for any A Є P(X)
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.4: Definition Of Function
Problem 55E
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Transcribed Image Text:Q1/a/ Let X be a topological space and A, B = P(X) show that whether the
statement is true or not int(A UB) = int(A) U int(B)
int(AUB):
b/ If ẞ is a collection of open sets in X Show that ẞ is a base for a topology o
X iff for each x Є X the collection ẞx = {BEẞ,x E B} is a nbhd base at x.
Q2/a/Let A be a subspace of a topological space X Show that:
1. ẞ is a base for X then {BNA, BE ẞ} is base for A.
2. HCA is open in A iff H = An U, U open in X.
b\Define the coarser topology and finer topology and give example with
prove: 1.The coarser topology than any topology 2. The finer topology than
any topology.
Q3/a/ If there exist an operation & on P(X) satisfying
1. Cl(0) = 0
2. A
Cl(A) for any A = P(X)
3. CI(CI(A)) = CI(A) for any A = P(X)
4. CI(AUB) = CI(A) U CI(B) for any A, B = P(X)
Show that there exist a unique topology T on X such that (A) = Cl(A) for
any A Є P(X)
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