Problem P7.3 This problem is about the metric space (X,d) from Problem P1.5 (the weighted Hamming distance). For every p N and every d;,....d, {0,1} we make the following notation: Ud,,....d,) == {T e X = (zM), 23 2% ), with W =dy, 2 =dy, ... 2P = b (In words: Uy, 4, is the set of all those # X which begin with the combination of components d;,...,d,. This notation has also appeared before, it was used in Problem P1.9.) (a) Prove that for every p N and every dy,...,d, {0,1}, the set Uy, 4 ) is an open subset of X. (b) Consider the collection of open sets U={U,, 4y | PEN, di,....dy 0,1} }. Prove that U is a basis of open sets for (X, d). () Consider the following statement: \"For every p N and every dy,....d, {0,1}, the set Uy, . q,) Is a closed subset of X.\" Is this statement true? Justify your answer (proof or counterexample)