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For $A=\mathbb{Q}\cap (0,1)$, I need to find the interior point, boundary points, limit points, isolated points, and whether its open, closed, or compact. Please check my work.

interior points: empty because of density of irrational between any two rationals.
boundary points: all real numbers in $(0,1)$ because of density of rational and irrationals
limit points: same as boundary point, because of density of rationals
isolated points: empty because $A$ minus its set of limit points is empty

Open: It's not open since A's boundary is not a subset of $\mathbb{R}$\A
Closed: It's not closed since A's boundary points are not subset of $A$
Compact It's not compact since it's not closed (but bounded)

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    $\begingroup$ Relative to $\mathbb R$ or $(0,1)?$ $\endgroup$ – spaceisdarkgreen Oct 29 '17 at 5:24
  • $\begingroup$ $A$ is a set of real numbers, so relative to $\mathbb{R}$ if I'm understanding your question correctly. $\endgroup$ – user496851 Oct 29 '17 at 5:26
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    $\begingroup$ @user496851 So you are asserting that $0$ and $1$ are not boundary points? $\endgroup$ – Lord Shark the Unknown Oct 29 '17 at 5:30
  • $\begingroup$ These questions are relative to a total topological space of which $A$ is a subset. Both $(0,1)$ and $\mathbb R$ would be natural choices here. $\endgroup$ – spaceisdarkgreen Oct 29 '17 at 5:30
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If the total space is $(0,1)$ your answers are all correct, although some of your justifications might need to be modified slightly for the last three.

If the total space is $\mathbb R,$ then the boundary and limit points are $[0,1],$ not $(0,1),$ since, e.g., every neighborhood of $0$ contains both points of $A$ and points of its complement.

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