Connected Set

[Bachelier]

I came across this question. We're looking for the conctd components of this set of circles: centered at (0,1) and with radius 1-1/n

B ((0,1), 1-1/n) for n = 3, ....to infinity

The radii are getting larger up to 1. I'm thinking the connectd comp. form an open set at infinity

would it be something like: \{(x,y) \in \mathbb{R}^2 | (x-1)^2+y^2<1/n \ with \ n \ being \ a \ large \ pos. \ integer \}

or is it the \emptyset

what do you think?

[HallsofIvy]

No, each B_n is a connectedness component and every connectedness component is one of the B_n.

[Bachelier]

thx,
I can see that the B_n are closed except when n approaches \infty. They never reach pt where radius = 0.
What then, are they open.

[HallsofIvy]

I don't know what you mean. Each Bn corresponds to a specific, finite, n. Limit as n goes to infinity of Bn is the circle with center at the origin and radius 1. No, they are not open. These circles are one dimensional subsets of R2. At each point on a circle, a small neighborhood will contain some points that are not on the circle so, far from being open, each Bn has empty interior.

[lavinia]

thx,
I can see that the B_n are closed except when n approaches \infty. They never reach pt where radius = 0.
What then, are they open.

In the subspace topology that the union of the circles inherits form the plane, each circle is both open and closed: open because it can be separated from the others by the intersection of two open sets in the plane; closed because every Cauchy sequence in it converges in it.