Which of the follwing subsets of R2 are complete metric spac

Which of the follwing subsets of |R^2 are complete metric spaces with the euclidean metric?

Which of the follwing subsets of R^2 are complete metric spaces with the euclidean metric? {(x, y) epsilon R^2 \\x^2 + y^2

Solution

(a) Not complete as it is not a closed subset of R² (which is complete)

(b) complete as it is a closed subset of R²

(c) Complete, as it is a closed subset of R²

(d) As f is continuous the set {f(x,y)=0} (being the inverse image of a closed set } is closed.

Hence complete