Question

For each of the following metric spaces (X, d) and subsets A S X decide whether A is open, closed, neither or both. You do not need to justify your answers. (a) X-Z, d is the discrete metric. ACX is any subset (b) X = R2, d = d2 is the Euclidean metric. A = {(x, yje R2 : x-y) (c) X = R2, d = d2 is the Euclidean metric. A (0, 1) × {0).

Answer 1

" x = Z, d is discrete metric Let A Subsetequalto x is any subset we know that In discrete metric space every singleton is open and arbitrary union of open sets is again open Therefore A Subsetequalto Z Rightarrow A = U_x elementof A {x} Here {x} is open Forall x elementof A Rightarrow A = U_x elementof A {x} is open - (*) Also, A' = complement of A A' is also subset of x = z which is again open (Because By (*)) Hence A is closed Rightarrow A Subsetequalto Z is open as well as closed X = R^2 d = d_2 is Euclidean metric A = {(x, y) elementof R^2 Therefore x = y} consider, f: R^2 rightarrow R as f (x, y) = x - y clearly f is continuous function Therefore f^-1 ({0}) = {(x, y) elementof R^2