B Dimension of subset containing two circles

1. May 11, 2017

RubinLicht

So Im reading a calculus book, and went online to find explanations for why a circle is 1D.
Theres the explanations that say something about zooming in very close and seeing that it's indistinguishable from a Real line.
Or you can specify any point on it with only one variable
Or if there was a train on the circle you can only go in two directions, forwards or back (this is a loose definition ofc)

My question: you have two concentric circles centered at the origin with different nonzero radii, is the subset of R2 as defined by these circles one dimensional or two dimensional?

Feel free to ask me for clarifications if something is wrong

2. May 11, 2017

Staff: Mentor

The circles themselves are one dimensional, their interior, exterior or the space between them is two dimensional.
The answers you found are all more or less correct. The essential part here is, how do you define the dimension? This will influence the kind of answer you will get. As a thumb rule, you define the dimension of something curved by the dimension of its tangent space at a certain point. Since the tangent of a circle is a line and not, e.g. a plane, it is one dimensional. The space between your two circles is two dimensional, because a tangent there would be a two dimensional plane.

3. May 11, 2017

RubinLicht

Clarification: I meant just the two curves. Not the space in between, but I see from your explanation that it is one dimensional. Thanks.

4. May 11, 2017

Staff: Mentor

For the set of two circles, you can use a generalization of the concept of dimensions, typically used for fractals. The result is 1 for every non-empty finite set of circles.

5. May 11, 2017