Cantor Space is Meager in Closed Unit Interval

Theorem
Let $\left({\mathcal C, \tau_d}\right)$ be the Cantor set considered as a topological subspace of the real number space $\R$ under the Euclidean topology $\tau_d$.

Then $\mathcal C$ is first category (meager) in $\left[{0 \,.\,.\, 1}\right]$.

Proof
From Cantor Space is Nowhere Dense, $\mathcal C$ is nowhere dense in $\left[{0 \,.\,.\, 1}\right]$.

So, trivially, $\mathcal C$ is the union of nowhere dense subsets of $\left[{0 \,.\,.\, 1}\right]$.

Hence the result from definition of first category.