Which sets are open, closed, or neither?

[calvino]

Determine with justification which sets are open, closed, or neither

i) {(x,y,z): x^2+ y^2 + z^2 +(xyz)^2 >= -1}

ii) {(x,y,z): x^2 + y^2 +z^2 >= 1}

iii) {(x,y,z): x^2- y- z >1}

iv) {(x,y): 3>= x^2- xy + y^2 >1

v) {(x,y): x^2 - y^2 >=0 }


So, my first insinct is to go about it using the definition of open sets. So I try and find a neighbourhood around a point in the set that is not completely contained in the set. What confuses me is that that method is not very definite. What if I can't find that neighbourhood?

Any help on how I should go about starting this question off?

 

[HallsofIvy]

I don't understand what you mean by "that method is not very definite". The definition is very definite!

However, you might find it easier to use a property that some text use as the definition: a set is open if and only if it contains none of its boundary points. "boundary" points may be difficult to define in general but with sets like you are given they are easy.

 

[calvino]

thanks.

now I'm just curious about what neither open nor closed means. Do you think you could explain to me a bit about that (or give an example?)?

 

[HallsofIvy]

A set is open if it contains none of its boundary points. A set is closed if it contains all of its boundary points. A set is neither open nor closed if it contains some but not all of its boundary points.

The set {x| 0<= x< 1} has "boundary" {0, 1}. It contains one of those but not the other and so is neither open nor closed.

For simple intervals like these, a set is open if it is defined entirely in terms of "<" or ">", closed if it is defined entirely in terms of "<=" or ">=", neither if it has both.

That is, however, for "simple intervals". Is the "set of all rational numbers between 0 and 1 (but not including 0 and 1)" open or closed (or neither)?

 

[calvino]

So in this case, by simply looking at the boundary points of the set, I come up with the following answers, in order.

closed, closed, open, neither, closed

Is this right, or do I have to consider manipulating the functions which make the set?