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 ECY, f-¹(Int(E)) ≤ Int(ƒ−¹(E)). (c) For any E CY, ƒ-¹(E) ≤ ƒ˜¹(E).

Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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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 ECY, f-¹(Int(E)) ≤ Int(ƒ−¹(E)).
(c) For any E CY, ƒ-¹(E) ≤ ƒ˜¹(E).
Transcribed Image Text: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 ECY, f-¹(Int(E)) ≤ Int(ƒ−¹(E)). (c) For any E CY, ƒ-¹(E) ≤ ƒ˜¹(E).
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