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Identify interior points, boundary points, limit points, and isolated points of:

$A = ((0,2) \cap \mathbb{Q} ) \cup ((1,3) \cap \mathbb{R} - \mathbb{Q}) \cup\mathbb{N}$

First the definitions:

If $\exists \delta > 0$ s.t. $(x-\delta , x+\delta) \subset A$, then x is said to be an interior point of A.

If, for every $\delta > 0$, the interval $(x-\delta , x+ \delta)$ contains a point in A and a point not in A, then x is a boundary point of A.

If, for every $\delta > 0$, the interval $(x-\delta, x+\delta)$ contains a point of A distinct from x, it is a limit point of A.

A point $x\in A$ is said to be an isolated point of A, if there is a $\delta > 0$ such that $(x - \delta, x + \delta) \cap A = \{x\}$.

My thoughts on the interior points are points inside the set that have an open neighborhood around them. It is hard to say with the given set because I have failed to interpret the set properly and my answer was $(0,3)$. Boundary points are points that have and open set with on point inside and one point outside for any interval, and again my interpretation of the set was wrong and i wrote $\{0\},\{3\}$. Limit points are points that have a subsequence that converges to a point inside the set, my answer was 1,2. Isolated points are points in the set that have no intervals between them, and my answer was $\emptyset$.

Advice given to me, was just be able to draw it out. My interpretation of the definitions my not be good enough to draw and my interpretation of the set was:

$(0,2) \in \mathbb{R}$ and $\mathbb{Q}$ or $(1,3) \in \mathbb{R}$ and $\mathbb{I}$ or $\mathbb{N}$

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Your interpretation of the set at the end of your post is correct. The key to solving this problem is:

$\star$any open interval centered at a point will contain both rational and irrational numbers.

Isolated points: By $\star$ the points in $[0,3]$ are not isolated. The set of isolated points is $\mathbb{N} - \{1,2,3\}$. To show this, just take $\delta = 1/2$.

Limit points: No point in $\mathbb{N} - \{1,2,3\}$ can be a limit point since these points are isolated. By $\star$ the set of limit points is $[0,3]$.

Interior points: In the interval $(0,2)$, only rational numbers are in $A$, and in the interval $(1,3)$, only irrational numbers are in $A$. Since the points in $\mathbb{N} - \{1,2,3\}$ are isolated they cannot be interior. So the set of interior points of $A$ is $$A_0 = (0,2) \cap (1,3) = (1,2).$$

Boundary points: Since all the points in $(1,2)$ are interior, they cannot be boundary points. Use $\star$ and the fact that the points in $\mathbb{N} - \{1,2,3\}$ are isolated to show that the set of boundary points is $$\partial A = ([0,1] \cap \mathbb{Q}) \cup ([2,3] \cap \mathbb{R} - \mathbb{Q}) \cup \mathbb{N} = A - (1,2).$$

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