# Determine the interior, the boundary and the closure of the set

• alexcc17
In summary: You should know it.In summary, the interior of the set is the set itself, the boundary is given by x^2-y^2=1, and there is no closure since the set is already open.
alexcc17

## Homework Statement

Determine the interior, the boundary and the closure of the set {z ε: Re(z2>1}
Is the interior of the set path-connected?

Re(z)=(z+z*)/2

## The Attempt at a Solution

Alright so z2=(x+iy)(x+iy)=x2+2ixy-y2

so Re(x2+2ixy-y2)= x2-y2 >1

So would the image be a hyperbola that starts when each axis is >1 leaving a hole in the center?

Boundary: {z ε: Re(z2=1}
Interior: none
Closure: {z ε: Re(z2>1}

Since there is no interior the question of interior path connectedness is mout.

I'm not sure if this is right though

alexcc17 said:

## Homework Statement

Determine the interior, the boundary and the closure of the set {z ε: Re(z2>1}
Is the interior of the set path-connected?

Re(z)=(z+z*)/2

## The Attempt at a Solution

Alright so z2=(x+iy)(x+iy)=x2+2ixy-y2

so Re(x2+2ixy-y2)= x2-y2 >1

So would the image be a hyperbola that starts when each axis is >1 leaving a hole in the center?

Boundary: {z ε: Re(z2=1}
Interior: none
Closure: {z ε: Re(z2>1}

Since there is no interior the question of interior path connectedness is mout.

I'm not sure if this is right though

##x^2-y^2=1## is a hyperbola. But that's not what you have. You've got ##x^2-y^2>1##. Can you figure out what that set is?

The set is given {z ε: Re(z^2)>1} so x^2 -y^2=52 or anything larger than 1 which is still a hyperbola.

alexcc17 said:
The set is given {z ε: Re(z^2)>1} so x^2 -y^2=52 or anything larger than 1 which is still a hyperbola.

You are thinking about it too hard. The boundary of your set is ##x^2-y^2=1##, sketch a graph of that hyperbola and then figure out where all of the points that satisfy ##x^2-y^2>1## lie relative to that hyperbola. Then answer the interior question again.

Last edited:
So you mean every point on that hyperbola excluding where each axis is 1,-1?

alexcc17 said:
So you mean every point on that hyperbola excluding where each axis is 1,-1?

No, the set of points that satisfy ##x^2-y^2>1## includes lots of points that aren't on the hyperbola ##x^2-y^2=1##. Rethink your answer about the interior.

Like this?

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alexcc17 said:
Like this?

Good job. Like that. What's the interior?

Would the interior be: {z ε: Re(z^2)>1} so the set itself, since the boundary is what it approaches and isn't actually part of the set.

alexcc17 said:
Would the interior be: {z ε: Re(z^2)>1} so the set itself, since the boundary is what it approaches and isn't actually part of the set.

Good enough, so now what's the boundary and what's the closure?

The boundary is {z ε: Re(z^2)=1} and there is no closure since it's an open set?

alexcc17 said:
The boundary is {z ε: Re(z^2)=1} and there is no closure since it's an open set?

You'd better look up the definition of closure again.

## 1. What is the interior of a set?

The interior of a set is the largest open subset of the set, meaning it contains all the points of the set that are not on the boundary. In other words, it is the set of all points that are surrounded by other points of the set.

## 2. How do you determine the boundary of a set?

The boundary of a set is the set of all points that are neither in the interior nor in the exterior of the set. In other words, it is the set of points that are on the edge of the set and are surrounded by both points inside and outside the set.

## 3. What is the closure of a set?

The closure of a set is the smallest closed subset of the set that contains all the points of the set. In other words, it is the set of all points that are either in the set or are limit points of the set.

## 4. How do you determine the interior of a set?

To determine the interior of a set, you can use the interior operator, denoted by int(A). This operator takes the union of all open sets contained in the set A.

## 5. Can a set have an empty interior or boundary?

Yes, a set can have an empty interior or boundary. This means that the set does not contain any points that are surrounded by other points or on the edge of the set. An example of this would be the set of all integers, which has an empty interior and boundary.

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