Sets of infinite Hausdorff dimension in a second countable metric space

Question

I am wondering if there exists an example of a second countable metric space $X$ containing a set $A$ with infinite Hausdorff dimension.

Answer 1

The infinite power $\mathbb{R}^{\mathbb{N}}$ is separable metrizable (hence second countable) and has infinite inductive dimension. This is a topologically defined notion, and is the smallest one among many related concepts; in particular, is less or equal to the Hausdorff dimension.

To see that $\mathrm{dim}\,\mathbb{R}^{\mathbb{N}} = \infty$: It can be proved that the inductive dimension of a space is at least equal to the sup of the dimensions of its subspaces (see Engelking, Dimension Theory, 1.1.2), and that $\mathbb{R}^{n}$ for $n\in{\mathbb{N}}$ has inductive dimension $n$.