Are the following sets open? Code Answer

Hello Developer, Hope you guys are doing great. Today at Tutorial Guruji Official website, we are sharing the answer of Are the following sets open? without wasting too much if your time.

The question is published on by Tutorial Guruji team.

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
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.