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).

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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