1. Let O1 and O2 be topologies on X. (1) Show that the identity map idX : (X, O1) → (X, O2) is continuous if O2 ⊂ O1. (2) Show that the identity map idX : (X, O1) → (X, O2) is not continuous if O1⊂≠O2.

1. Let O1 and O2 be topologies on X. (1) Show that the identity map idX : (X, O1) → (X, O2) is continuous if O2 ⊂ O1. (2) Show that the identity map idX : (X, O1) → (X, O2) is not continuous if O1⊂≠O2.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 1E: Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary...

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Question

1. Let O1 and O2 be topologies on X.

(1) Show that the identity map idX : (X, O1) → (X, O2) is continuous if O2 ⊂ O1.

(2) Show that the identity map idX : (X, O1) → (X, O2) is not continuous if O1⊂≠O2.

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