Analysis II | Part IB, 2004

Let XX and XX^{\prime} be metric spaces with metrics dd and dd^{\prime}. If u=(x,x)u=\left(x, x^{\prime}\right) and v=(y,y)v=\left(y, y^{\prime}\right) are any two points of X×XX \times X^{\prime}, prove that the formula

D(u,v)=max{d(x,y),d(x,y)}D(u, v)=\max \left\{d(x, y), d^{\prime}\left(x^{\prime}, y^{\prime}\right)\right\}

defines a metric on X×XX \times X^{\prime}. If X=XX=X^{\prime}, prove that the diagonal Δ\Delta of X×XX \times X is closed in X×XX \times X.

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