Homework Help: Elementary Topology

1. Sep 3, 2009

tracedinair

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. 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.

2. Sep 3, 2009

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?

3. Sep 3, 2009

aPhilosopher

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.

4. Sep 3, 2009

tracedinair

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

5. Sep 3, 2009

Dick

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