Is the projection from R^2 to R, with f(x,y)=x, both open and closed?

In summary, the conversation is about open and closed functions, specifically the projection from R^2 to R with the function f(x,y)=x. The person is confused because the Wikipedia article disagrees with their understanding that this function is both open and closed. Another person then provides an example where this function is not closed using the hyperbola xy=1. This clarifies the misunderstanding.
  • #1
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I'm trying to understand open and closed functions, and right now I'm on the projection from R^2 to R, with f(x,y)=x. It seems this is both open and closed, but the wikipedia article on open and closed functions seems to disagree:

(Note that product projections need not be closed. Consider for instance the projection p1 : R2 → R on the first component; A = {(x,1/x) : x≠0} is closed in R2, but p1(A) = R-{0} is not closed.)

I don't understand what exactly A is, and I can't think of any counterexamples myself. Are they talking about the same function as me? Can someone explain any of this?
 
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  • #2
In their example, A is the hyperbola given by the equation xy = 1.
 
  • #3
I see. So it's not closed. Thanks.
 

Related to Is the projection from R^2 to R, with f(x,y)=x, both open and closed?

What is an open function?

An open function is a mathematical term that refers to a function where the output value is not included in the range of possible values. This means that there is no specific value that the function cannot reach.

What is a closed function?

A closed function is a mathematical term that refers to a function where the output value is included in the range of possible values. This means that there is a specific value that the function cannot reach.

What is the difference between an open and closed function?

The main difference between an open and closed function is the inclusion or exclusion of the output value in the range of possible values. In an open function, the output value is not included, while in a closed function, the output value is included.

What are some examples of open functions?

Some examples of open functions include square root functions, logarithmic functions, and trigonometric functions. In these functions, there is no specific value that the function cannot reach, and thus they are considered open.

What are some examples of closed functions?

Some examples of closed functions include absolute value functions, polynomial functions, and linear functions. In these functions, there is a specific value that the function cannot reach, and thus they are considered closed.

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