# Determine the set of interior points, accumulation points, isolated points and boundary points.

I am working on Exercise $4.2.1$ and would like to know whether my answers are correct or not.

(a)

Interior Points:$\phi$ or the empty set

Accumulation Points:$\{0\}$

Isolated Points:$\{1,1/2,1/3...\}$

Boundary Points:$\{0,1\}$

(b)

Interior Points:$\{0\}$

Accumulation Points:$\{0\}$

Isolated Points:$\{1,1/2,1/3,...\}$

Boundary Points:$\{0,1\}$

(c)

Interior Points:$(0,1)\cup(1,2)\cup(2,3)..$ or the empty set

Accumulation Points:$[0,\infty)$

Isolated Points:$\phi$ or the empty set

Boundary Points:$\{0,\infty\}$

(d)

Interior Points: The entire set.

Accumulation Points: $[0,1]$

Isolated Points:$\phi$ or the empty set.

Boundary Points:$\{0,1\}$

(e)

Interior Points: $(1-\pi,1+\pi)$

Accumulation Points:$[\pi-1,\pi+1]$

Isolated Points:$\phi$ or the empty set.

Boundary Points:$\{1-\pi,1+\pi\}$

(f)

Interior Points: The entire set

Accumulation Points: The entire set along with $\sqrt 2$

Isolated Points:$\phi$ or the empty set.

Boundary Points:$\{\sqrt 2\}$

(g)

Interior Points: The entire set.

Accumulation Points:$\mathbb{R}$

Isolated Points:$\phi$ or the empty set

Boundary Points:$\phi$ or the empty set

(h)

Interior Points: $\phi$ or the empty set

Accumulation Points:$\mathbb{R}$

Isolated Points:$\phi$ or the empty set

Boundary Points:$\mathbb{Q}$

Sorry for the long post, but I would be really grateful if someone could verify my answers.

I assume the usual topology of the real line is being used for this question.

Your boundary points for $(a)$ are wrong.

Your interior and boundary points for $(b)$ are wrong.

For $(c)$, what is your choice of interior? One of them is right, and the other is wrong.
Also, the point at infinity is usually not considered a part of the real line.

If you agreed the entire set is open in case $(d)$, what made you doubt this in case $(c)$?

Your interior and boundary points for $(e)$ are wrong (maybe a typo here?).

Your accumulation and boundary points for $(f)$ are missing something.

Your boundary points for $(g)$ are wrong.

• Is boundary point for (a) $\{0\}$ and for (c) the interior is the entire set right? – model_checker Nov 17 '17 at 22:22
• Also the boundary points for (c) is the set $\{0,1,2,3,..\}$? Interior points for (e) should be $(\pi-1,\pi+1)$ and so the boundary points are $\{\pi-1,\pi+1\}.$ In (f) I am missing $-\sqrt{2}$ and so the accumlation points is $[-\sqrt{2},\sqrt{2}]$ and so the boundary points are $\{-\sqrt{2},\sqrt{2}\}.$ Finally for $(g)$ the boundary points are $\mathbb{R}.$ – model_checker Nov 17 '17 at 22:28
• Is this correct? Also thank you very much for replying! – model_checker Nov 17 '17 at 22:28
• For $(a)$ you're still off. For $(c)$, yes, the interior is the entire set, and you also got the boundary points right now. You've got $(e)$ and $(f)$ right this time. $(g)$ is still not right. – Fimpellizieri Nov 18 '17 at 5:37