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Exercise 3. Prove the following theorem: Theorem. Let f: X→ Y a map between the topological spaces X and Y. The following statements are all equivalent: 1) The map f is continuous. 2) For every open subset O of Y, the inverse image f-¹(O) is an open subset of X. 3) For every VE B, where B is a basis of open sets for Y, the inverse image f-¹(V) is an open subset of X. 4) For every closed subset F of Y, the inverse image f-(F) is a closed subset of X. 5) For every ACX, we have f(A) ≤ f(A).

Exercise 3. Prove the following theorem: Theorem. Let f: X→ Y a map between the topological spaces X and Y. The following statements are all equivalent: 1) The map f is continuous. 2) For every open subset O of Y, the inverse image f-¹(O) is an open subset of X. 3) For every VE B, where B is a basis of open sets for Y, the inverse image f-¹(V) is an open subset of X. 4) For every closed subset F of Y, the inverse image f-(F) is a closed subset of X. 5) For every ACX, we have f(A) ≤ f(A).

Elements Of Modern Algebra
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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#3 Need parts 1,2,3,4 and 5
Exercise 3. Prove the following theorem: Theorem. Let f: X→ Y a map between the topological spaces X and Y. The following statements are all equivalent: 1) The map f is continuous. 2) For every open subset O of Y, the inverse image f-¹(O) is an open subset of X. 3) For every VE B, where B is a basis of open sets for Y, the inverse image f-¹(V) is an open subset of X. 4) For every closed subset F of Y, the inverse image f-¹(F) is a closed subset of X. 5) For every ACX, we have f(A) ≤ f(A).
Transcribed Image Text:Exercise 3. Prove the following theorem: Theorem. Let f: X→ Y a map between the topological spaces X and Y. The following statements are all equivalent: 1) The map f is continuous. 2) For every open subset O of Y, the inverse image f-¹(O) is an open subset of X. 3) For every VE B, where B is a basis of open sets for Y, the inverse image f-¹(V) is an open subset of X. 4) For every closed subset F of Y, the inverse image f-¹(F) is a closed subset of X. 5) For every ACX, we have f(A) ≤ f(A).
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