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As part of an exercise, I want to calculate the interior, closure and boundary of the following sets in $\mathbb{R}^2$ (with the standard topology).
1. $\mathbb{Z} \times \mathbb{Z}$
2. $\mathbb{Q} \times (\mathbb{Q}\cap]0,+\infty[)$
I found the following solution and would like to verify whether those are correct.
1. Interior: $\emptyset$; Closure: $\mathbb{Z} \times \mathbb{Z}$; Boundary: $\mathbb{Z} \times \mathbb{Z}$
2. Interior: $\emptyset$; Closure: $\mathbb{R} \times (\mathbb{R}\cap[0,+\infty[)$; Boundary: $\mathbb{R} \times (\mathbb{R}\cap[0,+\infty[)$

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You are correct. You're basically done after interior and closure: the boundary follows easily as the difference between the closure and the interior. So empty interior implies boundary = closure, as we have here.

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