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- the usual metric space defined by d(x,y)= x-y prove the four propertis of metric spaceIs the set S = [0,1] with the discrete metric d separable? Explain.Prove that topological space E is not homeomorphic to the spaceY = {(x, y) ∈ E^2 : y = ± x} (E represents R equipped with Euclidean distance, E^2 represents R^2 equipped with euclidean distance)
- State true or false with a brief justification If the dual X' of a normed linear space X is fininte dimensional, then X is finite dimensionalSuppose that (X,d) is a metric space and R has the standard metric. Let f : X → R be a continuous function. For each r > 0, show that A={x∈X| |f(x)|<r} is open in X.Let (X,d) be a metric space. For x,y in X define e(x,y)=min{1,d(x,y)}. Prove that (X,e) is also a metric space.