is 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

It's a subset of the plane, so you can visualize it or draw a picture. What does it look like?

Quote:

---

Originally Posted by Tinyboss

It's a subset of the plane, so you can visualize it or draw a picture. What does it look like?

---

looks like a unit circle centred at origin

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?

Quote:

---

Originally Posted by Tinyboss

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?

---

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?

I don't know of a general way.

Quote:

---

Originally Posted by sorv1986

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?

---

Well, yes, but it isn't that easy to implement: a space X is NOT connected iff there is a

continuous surjection , with the second set with the inherited

topology from the reals (is thus a discrete space).

Tonio