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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 A FUA 1. Cl(0) = 0 VA 2. AC CL(A) for any AE P(X) 3. CI(CI(A)) = CL(A) for any A = P(X) 4. Cl(AUB) = CL(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 E P(X) b/ Show that ẞ is a base for a topology on X iff a. X=UBER B b. Let B1, B2 E ẞ with p E B₁n B₂ then there is a some B3 E ẞ with PE B3 CB₁n B₂

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 A FUA 1. Cl(0) = 0 VA 2. AC CL(A) for any AE P(X) 3. CI(CI(A)) = CL(A) for any A = P(X) 4. Cl(AUB) = CL(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 E P(X) b/ Show that ẞ is a base for a topology on X iff a. X=UBER B b. Let B1, B2 E ẞ with p E B₁n B₂ then there is a some B3 E ẞ with PE B3 CB₁n B₂

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 A FUA 1. Cl(0) = 0 VA 2. AC CL(A) for any AE P(X) 3. CI(CI(A)) = CL(A) for any A = P(X) 4. Cl(AUB) = CL(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 E P(X) b/ Show that ẞ is a base for a topology on X iff a. X=UBER B b. Let B1, B2 E ẞ with p E B₁n B₂ then there is a some B3 E ẞ with PE B3 CB₁n B₂
Transcribed Image Text: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 A FUA 1. Cl(0) = 0 VA 2. AC CL(A) for any AE P(X) 3. CI(CI(A)) = CL(A) for any A = P(X) 4. Cl(AUB) = CL(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 E P(X) b/ Show that ẞ is a base for a topology on X iff a. X=UBER B b. Let B1, B2 E ẞ with p E B₁n B₂ then there is a some B3 E ẞ with PE B3 CB₁n B₂
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