Function that is open but not closed?

[Unassuming]

Homework Statement

An example of a function that is open but not closed, and an example of a function that is closed but not open.

Homework Equations

The function f is open (closed) if f[A] is open (closed) for each open (closed) set A in X.

The Attempt at a Solution

This is a topology question and I don't understand it. The only attempt I have made was to guess at a function with < instead of =. If somebody could clarify on the point of what these "functions" look like. Are they the standard, f(x) = x^2 type, or something else?

 

[morphism]

They can be "standard." For example, if you give [itex]\mathbb{R}[/itex] its usual topology, then f(x)=x^2 (as a function from [itex]\mathbb{R}[/itex] to [itex]\mathbb{R}[/itex]) is closed but not open.

 

[Unassuming]

I was on the right track?

Is f(x) < x , open but not closed? Is any function with < or > open but not closed?

EDIT: I meant, given the usual topology and from R --> R

 

[HallsofIvy]

I have no idea what you mean by "f(x)< x" being "open but not closed". That is neither a function nor a set so "open" and "closed" do not apply.

It might be good for you to think about what a function is first. The problem asks for specific examples of functions.

 

[Unassuming]

I have no idea what you mean by "f(x)< x" being "open but not closed". That is neither a function nor a set so "open" and "closed" do not apply.

It might be good for you to think about what a function is first. The problem asks for specific examples of functions.

I have thought about what you said. Is y<x an open function?

EDIT: I am having trouble determining the difference between open, and open function.

 

[morphism]

I have thought about what you said. Is y<x an open function?

The real question is: is y<x a function?

Maybe a better question would be: what exactly do you mean by "y<x"?!

 

[Unassuming]

I don't feel that it is a function, but my book says that y^2 + x^2 < 9 is open. It doesn't specify that it is an open function though. I also don't have any other examples.

Is there a function that is open?

EDIT: I won't bother you guys (gals) too much more but here is my last attempt. The
function? f(x,y) = x^2 + y^2 < 9 . ??

 

[D H]

I don't feel that it is a function, but my book says that y^2 + x^2 < 9 is open.

That is not a function. It is, however, an open set.

Consider the simple function f(x)=x. The image of this function for any open set A is the open set A, and the image for any closed set B is the closed set B. f(x)=x is both an open and closed function. So this simple function doesn't work as far as solving your problem.

You need to come up with functions f and g where f is open but not closed and g is closed but not open. What this means is that f(A) is an open set for any open set A in the domain of f, but f(B) is not a closed set for any closed set in the domain of f, and similar for g.

 

[Unassuming]

What is it that makes f(x)=x^2 closed but not open? I do not understand how this is different from f(x)=x which is open and closed.

 

[D H]

What is it that makes f(x)=x^2 closed but not open? I do not understand how this is different from f(x)=x which is open and closed.

Given f(x)=x^2, what is the image of some open set that contains zero?