Finding Connected Components of a Set of Circles

[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 B_n corresponds to a specific, finite, n. Limit as n goes to infinity of B_n 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 B_n 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.