Answer Happy

1. Let X, D be a metric space. Suppose that 2 and y are two distinct points of X. Prove that there are open sets U and V in X such that u EU, YEV and Un V = $. (Hint: Let U = N(x, 4D(x, y))) 2. Determine which of the following subsets of the plane R2 with the Pytha- gorean metric are open. a) {(x, y)2 < 0) b) {(x, y) + y > 5} c) {(x, y)2 + y2 < 1, or (x, y) = (1, 0)} d) {(x, y) | < > 2 and y < 3} 3. Let p be any positive number. Prove that any D-p-neighborhood of R2 is D1-, D2-, and D3-open. 4. Prove that any subset of Rº which is D-open is Dj-open and, conversely, that any subset of the plane which is Di-open is D-open. 5. Make appropriate sketches for the proofs of (b) and (c) in Proposition 2. and n, are caid to be D