Complement of zero dimensional space

Conjecture. If S is a zero dimensional subspace of R^2,
then R^2 - S is a dense, path connected subspace.

An example is zero dimensional S = QxP \/ PxQ
and path connected R^2 - S = Q^2 x P^2. (P = R\Q)

Trivial examples are any countable S (regular countable
spaces are zero dimensional) and path connected R^2 - S.

A counter example to the converse is the set of all points
of rational slope lines, including the y-axis, through (0,0).

Counter examples are always welcome, because they preempt laboring over a proof.

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