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

[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?

Homework Equations

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

 

[Dick]

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?

Homework Equations

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?

 

[alexcc17]

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.

 

[Dick]

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.

 

[alexcc17]

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

 

[Dick]

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.

 

[alexcc17]

Like this?

 

[Dick]

Like this?

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

 

[alexcc17]

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.

 

[Dick]

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?

 

[alexcc17]

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

 

[Dick]

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.