4. Let E=]0,4[U]5, 6[ . (a) Prove that xo (b) Decide whether E is open, closed or compact, and determine the 3 is an interior point of E. • set E° of the interior points of E, • set ƏE of the boundary points of E , • set E' of the limit points of E and • the closure E of E.

(a) Let M R. Give the radius r and the center c of B(-2,5) n B(6, 7). (b) Let M = R. Using interval notations, give a simplified expression for the set L= [B(1,1) UB(5,1)] n B(3,2). (c) Let = 3(-1)" + and A = {,: ne N). Give the set W of accumulation points of A. (d) Let A₁-(-2n, 2n) and K = U An. Give a simplified expression for the interior Kº of K. NEN (e) Give the set T of isolated points of B(0, 1) U (3) U (2,5,7) (f) Let [P,Q] be a segment in R2 with midpoint H, and let (1,0) and (2, 1) be the components of the points P and H, respectively. Give the components (x, y) of the point Q. (g) Suppose that f: (M, d) →R satisfies d(f(x), f(y)) - vl. Let 21 e M and set n+1 = f(n), for n e N". Assume that M is complete and let a = lim zn. Give a simple formula satisfied by a.

3. Show that is a closed subset of R². {(x, y) = R² : y = x²}

Show that the linear map (x, y) H (x+ y, x- y) from (R?, || - |I,) to (R?, || - ||) is an isometry.