Is any compact metric totally disconnected space homeomorphic to a compact subspace of a Cantor space?

Question

Every compact metric totally disconnected perfect space is homeomorphic to a Cantor space.

Is every compact metric totally disconnected space homeomorphic to a compact subspace of a Cantor space?

In other words, if you have a compact metric totally disconnected space can you embed it in a compact metric totally disconnected perfect space?

Answer 1

Yes, first show that every zero-dimensional metric space is homeomorphic with a closed subset of the Baire space. Since the Baire space is a $G_\delta$ subset of the Cantor space, we have that every zero-dimensional metric space is a $G_\delta$ subspace of the Cantor space. Note that if our space was compact then the image of the homeomorphism is compact, as well.

The proof of this can be found in Kechris' Classical Descriptive Set Theory, Theorem 7.8.

Answer 2

From general facts we know that a compact totally disconnected Hausdorff space is zero-dimensional (has a base of clopen subsets) and as we also have a metric space, we have a countable clopen base for our space. The characteristic functions of these form an embedding family into $\{0,1\}^\omega$, which is (homeomorphic to) the Cantor set. In fact every compact Hausdorff totally disconnected space of weight $\kappa$ can be embedded into a Cantor cube of weight $\kappa$ in this way.