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'.

[1] L. E. J. Brouwer, Over de structuur der perfekte puntverzamelingen, *Amst. Ak. Versl.* **18** (1910) 833-842.