Solve Exercise 3.28 Definition 3.27. Let X be a set and T1 and T2 be topologies on X. If T1 C T2, we saythat Ti is coarser than T2 and that T2 is finer than T1. If T1 C Tz and T1 # T2, we say thatTi is strictly coarser than T2 and that T2 is strictly finer than T1.Exercise 3.28. It is obvious that if T1 is both coarser and finer than T2 on X, then thetopologies are equal (we say the topologies on X are equivalent). Prove it anyway. Thengive a reason why we might have chosen the "coarser/finer" language.

Definition: The topology τ1 is finer than τ2 on X, if τ2⊂τ1. Definition: The topology τ1 is coarser…

Exercise 3.28. It is obvious that if T1 is both coarser and finer than T2 on X, then the topologies are equal (we say the topologies on X are equivalent). Prove it anyway. Then give a reason why we might have chosen the "coarser/finer" language.

Exercise 3.28. It is obvious that if T1 is both coarser and finer than T2 on X, then the topologies are equal (we say the topologies on X are equivalent). Prove it anyway. Then give a reason why we might have chosen the "coarser/finer" language.

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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Question

Solve Exercise 3.28

Definition 3.27. Let X be a set and T1 and T2 be topologies on X. If T1 C T2, we say
that Ti is coarser than T2 and that T2 is finer than T1. If T1 C Tz and T1 # T2, we say that
Ti is strictly coarser than T2 and that T2 is strictly finer than T1.
Exercise 3.28. It is obvious that if T1 is both coarser and finer than T2 on X, then the
topologies are equal (we say the topologies on X are equivalent). Prove it anyway. Then
give a reason why we might have chosen the "coarser/finer" language.

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