Elementary Topology Homework: Boundaries of (x,y) on y = |x-2| + 3 - x

[tracedinair]

Homework Statement

Determine if the set of points (x,y) on y = |x-2| + 3 - x are bounded/unbounded, closed/open, connected/disconnect and what it's boundary consist of.

Homework Equations

The Attempt at a Solution

I know that the set is closed, and then by definition of a closed set it's boundary is itself. As far as bounded/unbounded goes, it seems unbounded when I graph it because I cannot see the entire graph. I'm unsure about connectedness and do not know how to determine it.

Any help is appreciated.

 

[Dick]

Well, in (x,y) x is certainly not bounded. It runs from -infinity to infinity. To think about connectedness, do you see that the function is continuous? That means its graph is a continuous curve with no breaks in it, right? What's the definition of 'connected' that you are using?

 

[aPhilosopher]

I know that the set is closed, and then by definition of a closed set it's boundary is itself.

Just so you know, that's not true in general. A closed set contains its boundary. For example, the unit 2-ball $$(x^{2} + y^{2})^{1/2} \leq 1$$ is closed but is not the same as its boundary which is the 1-sphere $$(x^{2} + y^{2})^{1/2} = 1$$. I could just be being pedantic though and you could well have known that and just not felt like spelling it out.

 

[tracedinair]

My guess then would that it is disconnected because of the absolute value in the function.

 

[Dick]

My guess then would that it is disconnected because of the absolute value in the function.

Guess?? Why are you guessing?? I'll ask you once more. What's the definition of a connected set?