For each of the following statements, give an example (if any) or, if not, write "none". In this problem X and Y represe

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For Each Of The Following Statements Give An Example If Any Or If Not Write None In This Problem X And Y Represe 1 (40.91 KiB) Viewed 23 times

For each of the following statements, give an example (if any) or, if not, write "none". In this problem X and Y represent metric spaces. (a) The function f : X Y is continuous, UC Y opens so that f^-1(U) does not open. (b) Cauchy sequence (xn) in R³ so that (xn) diverges. (c) The set SC X such that every point of S is a limit point of S. (d) The sequence (xn) converges on X, but is not a Cauchy sequence. (e) The set K CX is closed and finite, but not compact. (f) The set K≤ X is compact, but not closed. (g) The sets A and B are closed in X, but A U B is not. (h) The set A CX such that int (A) = A (i) The function f : X R is not continuous so that g(x) = f(x)² is continuous. (i) The sequence (xn) converges to xE A CX, but x is not a limit point of A.