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Finding the interior and the closure of a set

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Please see attached file. Please help write a rigorous proof on the first problem, which is called the EASY problem.

Consider the set A in the attached file. Compute, for the euclidean metric, the interior and the closure of A in R2. If you consider A as a subspace, is it complete?

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OK, let's carefully write the relevant definitions here.

x is an interior point of A if x belongs to A and there is an open ball B(x,e)={y in R^2: d(x,y)<e} that is contained in A.
x belongs to the closure of A, if for any open ball B(x,e)={y in R^2: d(x,y)<e}, the intersection of the ball with A is not empty.

Now, let A be as in your example. A={(x_1,x_2) in R^2: x_1 = 0 or x_2 = 0}. The Euclidean metric is d(x,y)=sqrt([x_1-y_1]^2+[x_2-y_2]^2).
I will claim that (1) the interior of A is empty, and that the (2) closure of A is A.

(1) ...

Solution Summary

We find the interior and the closure of the given subset of R^2. Detailed exaplanations are given.