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

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SUMMARY

The projection function from R² to R defined by f(x,y) = x is both open and closed under specific conditions. However, the projection of certain closed sets, such as A = {(x, 1/x) : x≠0}, can result in a non-closed set in R, specifically R - {0}. This demonstrates that while the function can be open and closed, it does not universally apply to all subsets of R². The hyperbola defined by the equation xy = 1 serves as a counterexample to the claim of closedness in projections.

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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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In their example, A is the hyperbola given by the equation xy = 1.
 
I see. So it's not closed. Thanks.
 

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