# Does this transfinitely defined set have empty interior?

1. Oct 10, 2013

### lugita15

Let us use the axiom of choice to well-order the set of open intervals of R, and suppose this well-ordering has order type gamma. For any ordinal alpha less than gamma, let I_alpha be the alpha'th interval according to the well-ordering.

Now let us use transfinite recursion to define a set of points in R. Let a_1 be any point in I_1. For any ordinal alpha less than gamma, define a_alpha as follows: if any of the a_beta's for beta less than alpha are in I_alpha, let a_alpha be the least such a_beta. Otherwise, let a_alpha just be any element of I_alpha.

Now if we take the set A of all a_alpha for all alpha less than gamma, then clearly A is dense in R, since it intersects every open interval by construction. But my question is, is it possible for A to have a nonempty interior, i.e. is it possible for A to contain any open intervals? I'd like to show that it's impossible, and I'd like to generalize this to arbitrary metric spaces.

Any help would be greatly appreciated.