Brouwer proved that a topological space is homeomorphic to the Cantor set if and only if
1) it's non-empty,
2) it's compact,
3) it's totally disconnected,
4) it has no isolated points, and
5) it's metrizable.
What are some nice examples of spaces that meet conditions 1)-4) but not 5)?
Perhaps it's worth noting that a compact Hausdorff space is metrizable if and only if it's second-countable, by a spinoff of the Urysohn Metrization Theorem. I suppose I'd prefer my examples to be Hausdorff. If so, they must be non-second-countable. Such spaces might thus be seen as 'like the Cantor set, but bigger'.
 L. E. J. Brouwer, Over de structuur der perfekte puntverzamelingen, Amst. Ak. Versl. 18 (1910) 833-842.