Chapter 7.2, Problem 4E

(a)

To determine

To find: The set of interior points for the given subset of $ℝ$ .

Answer to Problem 4E

The interior of the given set is $(-1,1)$ .

Explanation of Solution

Given Information:

The subset that is given is $(-1,1)$

The interior of the given set is $(-1,1)$ as every $x\in (-1,1)$ we are able to find $\delta >0$ where $\delta =\min \{x+1,1-x\}$ so that $N(x,\delta )\in (-1,1)$ .

Thus, the given set has an interior point.

Hence the interior of the given set is $(-1,1)$ .

(b)

To determine

To find: The set of interior points for the given subset of $ℝ$ .

Answer to Problem 4E

The interior of the given set is $(-1,1)$ .

Explanation of Solution

Given Information:

The subset that is given is $(-1,1]$

We know that $(-1,1)\subseteq (-1,1]$ if 1 being the boundary point but its not interior point since $x\in N(1,\delta )$ so that $x\notin (-1,1]$ .

Thus, the interior point is $(-1,1)$ .

Hence, the interior of the given set is $(-1,1)$ .

(c)

To determine

To find: The set of interior points for the given subset of $ℝ$ .

Answer to Problem 4E

The interior point of the given set is $\varphi$ .

Explanation of Solution

Given Information:

The subset that is given is $\mathbb{Q}$ .

  $\mathbb{Q}$ do not have interior point because any neighborhood of $x\in \mathbb{Q}$ has an irrational point.

Thus, the given set do not have any interior point.

Hence, the interior point of the given set is $\varphi$ .

(d)

To determine

To find: The set of interior points for the given subset of $ℝ$ .

Answer to Problem 4E

The interior point of the given set is $\varphi$ .

Explanation of Solution

Given Information:

The subset that is given is $ℝ-\mathbb{Q}$ .

  $ℝ-\mathbb{Q}$ do not have interior point because all the neighborhood contains rational points

It is dense in $ℝ$ as its both rational and irrational.

Thus, it has no interior point.

Hence, the interior point of the given set is $\varphi$ .

(e)

To determine

To find: The set of interior points for the given subset of $ℝ$ .

Answer to Problem 4E

The interior point of the given set is $\varphi$ .

Explanation of Solution

Given Information:

The subset that is given is $\{\frac{1}{3k}:k\in \mathbb{N}\}$

Let us take any arbitrary $x=\frac{1}{3k}$ and select $0<\frac{1}{3k}-\frac{1}{3k+3}$ . Then the neighborhood point $N(x,\delta )$ will contain only a single point $\{\frac{1}{3k}:k\in \mathbb{N}\}$ . That is $x$ itself therefore $x$ cannot be an interior point of the set given.

Thus, there is no interior point of the given set.

Hence, the interior point of the given set is $\varphi$ .

(f)

To determine

To find: The set of interior points for the given subset of $ℝ$ .

Answer to Problem 4E

The interior point of the given set is $\varphi$ .

Explanation of Solution

Given Information:

The subset that is given is $\{\frac{1}{3k}:k\in \mathbb{N}\}\cup \left\{0\right\}$ .

The given set do not have any interior point because for any point of $\delta >0$ , $N(0,\delta )$ which the neighborhood will contain negative numbers which doesn’t contain in the given set.

Thus, there is no interior point of the given set.

Hence, the interior point of the given set is $\varphi$ .

(g)

To determine

To find: The set of interior points for the given subset of $ℝ$ .

Answer to Problem 4E

The interior point of the given set is $ℝ-\mathbb{N}$ .

Explanation of Solution

Given Information:

The subset that is given is $ℝ-\mathbb{N}$ .

If we take any number $x$ in the set given and select $\delta =\inf \{x-n:n\in \mathbb{N}\}$ then $N(x,\delta )$ lies on 0 $ℝ-\mathbb{N}$ entirely.

Thus, there is interior point in the given set.

Hence, the interior point of the given set is $ℝ-\mathbb{N}$ itself.

(h)

To determine

To find: The set of interior points for the given subset of $ℝ$ .

Answer to Problem 4E

The interior point of the given set is $ℝ-\{\frac{1}{3k}:k\in \mathbb{N}\}-\left\{0\right\}$ .

Explanation of Solution

Given Information:

The subset that is given is $ℝ-\{\frac{1}{3k}:k\in \mathbb{N}\}$ .

The point $\frac{1}{3k}$ will converge to 0. Now let us take any point $x\ne 0$ so now we select $\delta =\inf \{x-\frac{1}{3k}:k\in \mathbb{N}\}$ such that $N(x,\delta )\subseteq ℝ-\{\frac{1}{3k}:k\in \mathbb{N}\}$ . This tells us that for every point $x\ne 0$ of the set given is an interior point. The reason behind $x=0$ not being an interior point is that $N(0,\delta )$ and it contains many infinite points of $\{\frac{1}{3k}:k\in \mathbb{N}\}$ . Specially for all values of $k\ge n\in \mathbb{N}$ for $\frac{1}{3k}<\delta$ .

Thus, the given set has an interior point.

Hence, the interior point of the given set is $ℝ-\{\frac{1}{3k}:k\in \mathbb{N}\}-\left\{0\right\}$ .

(i)

To determine

To find: The set of interior points for the given subset of $ℝ$ .

Answer to Problem 4E

The interior point of the given set is ${\displaystyle \underset{n\in \mathbb{N}}{\cup }(n+0.1,n+0.2)}$ .

Explanation of Solution

Given Information:

The subset that is given is ${\displaystyle \underset{n\in \mathbb{N}}{\cup }(n+0.1,n+0.2)}$ .

The interior point of the set given is $(n+0.1,n+0.2)$ because the interior point of the interval $(n+0.1,n+0.2)$ is $(n+0.1,n+0.2)$ itself.

Thus, the given set has an interior point.

Hence, the interior point of the given set is ${\displaystyle \underset{n\in \mathbb{N}}{\cup }(n+0.1,n+0.2)}$ .

(j)

To determine

To find: The set of interior points for the given subset of $ℝ$ .

Answer to Problem 4E

The interior point of the given set is $\varphi$ .

Explanation of Solution

Given Information:

The subset that is given is $\mathbb{Z}$ .

If we select $0<\delta <1$ then $N(x,\delta )$ will contain only a single point on $\mathbb{Z}$ which is $x$ and it is not an interior point.

Thus, the given set do not have any interior point.

Hence, the interior point of the given set is $\varphi$ .