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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