# Open/Closed Set Homework: Examples of \mathbb{R}^2

• Ted123
In summary, the conversation discussed the concept of a set being closed in \mathbb{R}^2 and provided examples of sets that may or may not be closed. It was determined that the set \{ (f(x),g(x)) : x\in\mathbb{R} \} can be either closed or not closed depending on the functions f and g. It was also clarified that being open and being not closed are two distinct concepts. One example of a function from \mathbb{R} to \mathbb{R} that is not closed was given as g(x)=e^x, which results in the set \{ (0,e^x) : x\in\mathbb{R}
Ted123

## Homework Statement

If $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ are continuous functions, give examples to show that the set $\{ (f(x),g(x)) : x\in\mathbb{R} \}$ might or might not be closed in $\mathbb{R}^2$.

## The Attempt at a Solution

Letting $f(x)=g(x)=0$ gives the set equal to $\{ (0,0) \}$, a singleton and singleton sets are closed.

What functions would make the set open?

That's a pretty trivial example. You should probably think about it a little more before someone just gives you the answer. Suppose f(x) and g(x) have a horizontal asymptote as x->inf. There's other ways it can fail to be closed as well, but that should get you started.

Dick said:
That's a pretty trivial example. You should probably think about it a little more before someone just gives you the answer. Suppose f(x) and g(x) have a horizontal asymptote as x->inf. There's other ways it can fail to be closed as well, but that should get you started.

Well if the functions have horizontal asymptotes they won't be continuous on the real line for a start.

If we take f(x) a constant function, what condition must g(x) satisfy in order that the set isn't closed?

Ted123 said:
What functions would make the set open?

One VERY IMPORTANT THING:

Open is NOT the same as "not closed".

You don't need to find a set that is open, you need to find a set that is not closed. These are completely different questions!

Ted123 said:
Well if the functions have horizontal asymptotes they won't be continuous on the real line for a start.

If we take f(x) a constant function, what condition must g(x) satisfy in order that the set isn't closed?

I said HORIZONTAL asymptote. Why do you think that means it wouldn't be continuous?

Sorry, of course it can be continuous.

Is the set $\{ (0,x):x\in\mathbb{R} \}$ not closed?

Ted123 said:
Sorry, of course it can be continuous.

Is the set $\{ (0,x):x\in\mathbb{R} \}$ not closed?

It doesn't look closed to me. What do you think? Keep micromass's comment in mind too. A set that is 'not closed' doesn't have to be open.

Dick said:
It doesn't look closed to me. What do you think? Keep micromass's comment in mind too. A set that is 'not closed' doesn't have to be open.

I changed my mind. I thought you were trying to write an open subsegment of a straight line. You're aren't, are you? Your set is the x-axis. It's closed in R^2. How would you show that?

Dick said:
I changed my mind. I thought you were trying to write an open subsegment of a straight line. You're aren't, are you? Your set is the x-axis. It's closed in R^2. How would you show that?

I've just realized that I'm confusing 'open' and 'not closed'.

With $f$ a constant function, I need to find $g$ such that $g(\mathbb{R})$ is not closed. What is an example a function from R to R that isn't closed?

Ted123 said:
I've just realized that I'm confusing 'open' and 'not closed'.

With $f$ a constant function, I need to find $g$ such that $g(\mathbb{R})$ is not closed. What is an example a function from R to R that isn't closed?

So you aren't going to try the horizontal asymptote suggestion?

Dick said:
So you aren't going to try the horizontal asymptote suggestion?

$g(x)=e^x$.

Then $\{(0,e^x) :x\in\mathbb{R} \}$ is not closed is it?

Ted123 said:
$g(x)=e^x$.

Then $\{(0,e^x) :x\in\mathbb{R} \}$ is not closed is it?

No, it isn't. Why isn't it closed?

Dick said:
No, it isn't. Why isn't it closed?

The boundary of $A = \{(0,e^x) : x\in \mathbb{R} \}$ is $\partial A =\{(0,0)\}$ and $(0,0) \notin A$ so $A$ doesn't contain all its boundary and is therefore closed.

Ted123 said:
The boundary of $A = \{(0,e^x) : x\in \mathbb{R} \}$ is $\partial A =\{(0,0)\}$ and $(0,0) \notin A$ so $A$ doesn't contain all its boundary and is therefore closed.

Good answer! Except that the boundary of A is actually all of A AND (0,0). Every point in A is also a boundary point of A. But the critical thing is that (0,0) is on the boundary of A but not in A.

Last edited:
Would you express the boundary as $\partial A = A\cap \{(0,0)\}$?

Ted123 said:
Would you express the boundary as $\partial A = A\cap \{(0,0)\}$?

Certainly not, it's a union not an intersection.

## 1. What is an open set in \mathbb{R}^2?

An open set in \mathbb{R}^2 is a set of points in the two-dimensional coordinate plane that does not include its boundary. This means that any point within the set can be surrounded by an open ball (a set of points within a certain radius) that is also contained within the set.

## 2. How is a closed set different from an open set in \mathbb{R}^2?

A closed set in \mathbb{R}^2 is a set of points in the two-dimensional coordinate plane that includes its boundary. This means that the boundary points are also considered part of the set. Unlike an open set, a closed set cannot be surrounded by an open ball without including points outside of the set.

## 3. Can you provide an example of an open set in \mathbb{R}^2?

One example of an open set in \mathbb{R}^2 is the set of all points within a circle of radius 1 centered at the origin (0,0). This set does not include its boundary, which is the circumference of the circle.

## 4. How do you determine if a set is open or closed in \mathbb{R}^2?

A set in \mathbb{R}^2 is considered open if it does not include its boundary, and it is considered closed if it does include its boundary. To determine this, you can visualize the set in the coordinate plane and see if the points on the boundary are part of the set or not.

## 5. Are there sets that can be both open and closed in \mathbb{R}^2?

In \mathbb{R}^2, a set can only be either open or closed, but not both. However, in other mathematical spaces, such as \mathbb{R}^n, a set can be both open and closed, known as a clopen set.

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