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.