(a) Prove that if X is path-connected and f: X Y is continuous, then the image...

 

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(a) Prove that if X is path-connected and f: X→ Y is continuous, then the image f(X) is path-connected. (b) Prove that path-connectedness is a topological property, i.e. if X and Y are homeomorphic topological spaces, then X is path-connected if and only if Y is path-connected. Hint: Use (a) and note that it suffices to show one direction, then reverse the names X → Y. (a) Prove that if X is path-connected and f: X→ Y is continuous, then the image f(X) is path-connected. (b) Prove that path-connectedness is a topological property, i.e. if X and Y are homeomorphic topological spaces, then X is path-connected if and only if Y is path-connected. Hint: Use (a) and note that it suffices to show one direction, then reverse the names X → Y.

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a of x is path connected and fixy is centinuens then fx is path connected Proof so x f... View the full answer