Chapter 16.3, Problem 34E

To determine

(a)

Whether the given set is open or not

Solution:

Open

Explanation:

1) Concept:

$D$ is open means that for every point $P$ in $D$ there is a disk with center $P$ that lies entirely in $D.$

2) Given:

$\left\{\left(x,y\right)|\left(x,y\right)\ne \left(\mathrm{2,3}\right)\right\}$

3) Calculations:

Consider the given set,

$D=\left\{\left(x,y\right)|\left(x,y\right)\ne \left(\mathrm{2,3}\right)\right\}$

Sketch of the region $D$ is

The set $D=\left\{\left(x,y\right)|\left(x,y\right)\ne \left(\mathrm{2,3}\right)\right\}$ consists of all points in the $xy-$ plane except for $\left(\mathrm{2,3}\right)$

That is $D={\mathbb{R}}^2-\left\{\left(\mathrm{2,3}\right)\right\}$

For points $P$ not on the boundary one has always draw a disk with center at $P$ which lies completely in the set.

The set $D$ has only one boundary point, namely $\left(\mathrm{2,3}\right),$ which is not included in the set.

By using the concept, the set $D$ is open.

Conclusion:

The set $D$ is open.

(b)

Whether the given set is connected or not

Solution:

Connected

Explanation:

1) Concept:

$D$ is connected means that any two points in $D$ can be joined by a path that lies entirely in $D.$

2) Given:

$\left\{\left(x,y\right)|\left(x,y\right)\ne \left(\mathrm{2,3}\right)\right\}$

3) Calculations:

Consider the given set,

$D=\left\{\left(x,y\right)|\left(x,y\right)\ne \left(\mathrm{2,3}\right)\right\}$

Sketch of the region $D$ is

The set $D=\left\{\left(x,y\right)|\left(x,y\right)\ne \left(\mathrm{2,3}\right)\right\}$ consists of all points in the $xy-$ plane except for $\left(\mathrm{2,3}\right)$

That is $D={\mathbb{R}}^2-\left\{\left(\mathrm{2,3}\right)\right\}$

The region $D$ consists of only one piece.

Also, any two points chosen in $D$ can always joined by a path that lies entirely in $D.$

By using the concept, the set $D$ is connected.

Conclusion:

The set $D$ is connected.

(c)

Whether the given set is simply connected or not.

Solution:

Not simply connected

Explanation:

1) Concept:

A simply connected region in the plane is a connected region $D$ such that every simple closed curve in $D$ enclosed only points that are lies in $D.$ In other words, the region cannot have a hole and cannot be in two pieces.

2) Given:

$\left\{\left(x,y\right)|\left(x,y\right)\ne \left(\mathrm{2,3}\right)\right\}$

3) Calculations:

Consider the given set,

$D=\left\{\left(x,y\right)|\left(x,y\right)\ne \left(\mathrm{2,3}\right)\right\}$

Sketch of the region $D$ is

The set $D=\left\{\left(x,y\right)|\left(x,y\right)\ne \left(\mathrm{2,3}\right)\right\}$ consists of all points in the $xy-$ plane except for $\left(\mathrm{2,3}\right)$

That is $D={\mathbb{R}}^2-\left\{\left(\mathrm{2,3}\right)\right\}$

The region $D$ is not simply connected, it has a hole at $\left(\mathrm{2,3}\right).$

Consider two points $A\left(\mathrm{2,1}\right)$ and $B\left(\mathrm{2,4}\right),$ which belongs to the set $D,$ marked in the graph above.

See that the path joining the two points $A$ and $B$ does not lie completely inside the given set.

Since it passes through the point $\left(\mathrm{2,3}\right)$ which is not in the set.

Conclusion:

The set $D$ is not simply connected.