# Thread: Independent of path; conservative vector field

1. ## Independent of path; conservative vector field

Hi,

I need your help for the below concepts.

1) A region D is OPEN if it doesn't contain any of its boundary points.
Can you give me an example of an OPEN region?

Can you also give me an example of open, simply connected region?

Thank you very much.

2. Originally Posted by kittycat
Hi,

I need your help for the below concepts.

1) A region D is OPEN if it doesn't contain any of its boundary points.
Can you give me an example of an OPEN region?

Can you also give me an example of open, simply connected region?

Thank you very much.
I believe that the interval of the real line (0, 1) satisfies both requirements.

If I am right, then the circular region of the plane (the "open unit disc") $\displaystyle A = \{ (r, \theta) | 0 < r < 1, \theta ~\text{anything } \}$ will be another example of both.

For open and not simply connected: $\displaystyle (-1, 0) \cup (0, 1)$. (I don't think I can do open and connected, though I'm sure they exist.)

(These examples are taken using the usual topology on the real line and $\displaystyle \mathbb{R}^2$, of course.)

-Dan

3. i read your topic title.. i think you are to consider paths here..

the region enclosed by $\displaystyle C_1: y=x^2$ and $\displaystyle C_2: y=2x$ from (0,0) to (2,4) to (0,0) again, taking C_1 to C_2 but does not include (0,0) is simply connected which is open.

$\displaystyle D = {(x,y) | x^2 + y^2 < 1}$ is open..

4. Here some unbounded examples.
The first quadrant, x>0 & y>0, is connected open set.
The upper half-plane, y>0, is connected open set.
The vertical plank, -2<x<2, is connected open set.