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.

## Best Answer

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.