Are the following sets open? Code Answer

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Question: Are the following sets open? Provide reasons for your answers.

  1. $A = {(x,y): x^2 + y^2 le1 }$ in $mathbb{R^2}$
  2. $A = {(x,y): 0 leq y <1 }$ in $mathbb{R^2}$
  3. $A = { (x,y,z) : z>0}$ in $mathbb{R^3}$
  4. $A = {(x,y,z) : x=y=z }$ in $mathbb{R^3}$
  5. $A = {(x,y): x geq 0 }$ in $mathbb{R^2}$
  6. $A = { (x,y): x>0 }$ in $mathbb{R^2}$
  7. $[1,2]$ in $mathbb{R}$

My Attempts:

  1. $A^0 = {(x,y): x^2 + y^2 < 1 } neq A implies A$ not open
  2. $A^0 = { (x,y): 0 < y < 1 } neq A implies A$ not open
  3. $A^0 = { (x,y,z) : z>0 } = A implies A$ is open
  4. (I am not sure about this one at all)
  5. $A^0 = {(x,y): x >0} neq A implies A$ not open
  6. $A^0 = { (x,y): x>0 } = A implies A$ is open
  7. $([1,2])^0 = (1,2) neq [1,2] implies [1,2]$ not open

Are these correct? Also, any hints about the ones I am unsure about will be much appreciated :).

I am reading through some notes on open sets and came across these examples, which I attempted to test how well my understanding of open sets is.

Answer

Yes, these are correct. As for 4., keep in mind that $A$ is a line in 3-space. It shouldn’t be to hard to see that $A^o=emptyset$.

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