I am compiling a sort of list of interior, isolated, boundary and limit points of different number sets for my knowledge. will deeply appreciate your input on this.
$\mathbb{N}$ - Set of Natural Numbers
- Interior Point
- Isolated Point
- Boundary Point
- Limit Point
$\mathbb{Q}$ - Set of Rational Numbers
- Interior Point
- Isolated Point
- Boundary Point
- Limit Point
$\mathbb{P}$ - Set of Irrational Numbers
- Interior Point
- Isolated Point
- Boundary Point
- Limit Point
$\mathbb{R}$ - Set of Real Numbers
- Interior Point
- Isolated Point
- Boundary Point
- Limit Point
Thanks in Advance.
Sid
If we work within $\Bbb R$ with the usual topology; as a bonus I added topological characterisations of each as well:
$\Bbb N$ is closed, discrete in itself (so every point is isolated), but has no interior points in $\Bbb R$ so all its points are boundary points $$\partial \Bbb N = \overline{\Bbb N}\setminus \operatorname{int}(\Bbb N)=\Bbb N\setminus \emptyset= \Bbb N$$ and as a space in its own right it's the unique (up to homeomoerphism) countably infinite discrete space.
$\Bbb Q$ is dense, so $\overline{\Bbb Q}=\Bbb R$ and no open interval/ball of $\Bbb R$ only contains rationals; there are always irationals too, so there are no interior points and no point is isolated. So the boundary is $\Bbb R$: $$\partial \Bbb Q = \overline{\Bbb Q}\setminus \operatorname{int}(\Bbb Q)=\Bbb R\setminus \emptyset= \Bbb R$$ and as a space in its own right it's (by a classical theorem) the unique (up to homeomorphism) countable metrci space without isolated points.
$\Bbb P$ is also dense and for the same reason as $\Bbb Q$ has no interior points in $\Bbb R$ and also no isolated points (so all points are accumulation points of it) and the boundary is $\Bbb R$ again:$$\partial \Bbb P = \overline{\Bbb P}\setminus \operatorname{int}(\Bbb P)=\Bbb R\setminus \emptyset= \Bbb R$$ and as a space in its own right it's the unique (up to homeomorphism) completely metrisable separable metric space in which all compact sets have empty interior (nowhere locally compact).
$\Bbb R$ is easy: every point is interior, it equals its own closure, the boundary is empty, no isolated points. It's the unique (up to isomorphism) separable metric space that is connected, locally connected and such that each point is a strong cut point ($X\setminus \{p\}$ has exactly two components).