Compact subsets of {0,1}^N

Let K,L be two closed subsets of S = {0,1}^N without isolated points.

How to show K and L are homeomorphic?

{0,1} is understood to be a discrete subspace of R.
S is zero dimensional compact Hausdorff space.
K,L are uncountable zero dimensional compact subsets.

Yes S is homeomorphic to the Cantor set, however without reference
to the Cantor set and theorems about it, how can one show K and L
are homeomorphic directly from the properties of S?

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