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Let X and Y be topological spaces and let f : X → Y be a function. Prove that the following statements are equivalent: (a) f: XY is continuous. (b) For any E CY, f-¹(Int(E)) ≤ Int(f-¹(E)). (c) For any ECY, ƒ˜¹(E) ≤ f¯¹(E).

Let X and Y be topological spaces and let f : X → Y be a function. Prove that the following statements are equivalent: (a) f: XY is continuous. (b) For any E CY, f-¹(Int(E)) ≤ Int(f-¹(E)). (c) For any ECY, ƒ˜¹(E) ≤ f¯¹(E).

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 8E: If x and y are elements of an ordered integral domain D, prove the following inequalities. a....
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Let X and Y be topological spaces and let ƒ : X → Y be a function. Prove that the following statements are equivalent: (a) f: X→Y is continuous. (b) For any EC Y, ƒ−¹(Int(E)) ≤ Int(f-¹(E)). (c) For any ECY, ƒ−¹(E) ≤ ƒ˜¹(Ē).
Transcribed Image Text:Let X and Y be topological spaces and let ƒ : X → Y be a function. Prove that the following statements are equivalent: (a) f: X→Y is continuous. (b) For any EC Y, ƒ−¹(Int(E)) ≤ Int(f-¹(E)). (c) For any ECY, ƒ−¹(E) ≤ ƒ˜¹(Ē).
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