is the set A={ (x,y)∈R² : x²+ y² =1} connected? how do we prove it?

is there any general proof? or some quick method by which we can tell any set is connected or not

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- May 21st 2011, 12:21 PMsorv1986is the set A={ (x,y)∈R² : x²+ y² =1} connected?
is the set A={ (x,y)∈R² : x²+ y² =1} connected? how do we prove it?

is there any general proof? or some quick method by which we can tell any set is connected or not - May 21st 2011, 12:26 PMTinyboss
It's a subset of the plane, so you can visualize it or draw a picture. What does it look like?

- May 21st 2011, 12:27 PMsorv1986
- May 21st 2011, 12:50 PMTinyboss
And is that connected, under the subspace topology inherited from R^2? Here's a hint: a continuous image of a connected set is connected. Can you express the unit circle as the continuous image of some set you know is connected?

- May 21st 2011, 01:11 PMsorv1986
If f : [0,1] -> R2 defined by f(t) = (cost, sint) then the map is continuous and the closed interval [0,1] is connected. So the image of f should be connected.and i think the image of f is exactly the unit circle centerd at origin. is it right?

but is there any general method for proving or disproving any subset of R2 connected? - May 21st 2011, 01:13 PMTinyboss
I don't know of a general way.

- May 21st 2011, 03:28 PMtonio