Almost-Cantor Set Example

Question

I'm in a Facebook group about sharing false "theorems" and one of them was the following:

Let X be a compact, metrizable and totally disconnected, then X is homeomorphic to the Cantor set.

I can come up with plenty of counter examples, but none with the same cardinality as the Cantor set. I've tried things like $C\times\{0,1\}$ for example, but that's still homeomorphic to the Cantor set.

Can you provide an example of a set that satisfies the hypothesis of the "theorem," is of the same cardinality as the cantor set, but isn't homeomorphic to it?

Answer 1

The Cantor set has no isolated points, so add one. The result is still compact, metrizable, and totally disconnected.