Cantor Space is not Locally Connected

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 not locally connected.

Proof
Let $\mathcal B$ be a basis of $\mathcal C$.

Let $A \in \mathcal B$.

By definition of $\mathcal B$, $A$ is an open set.

But the Cantor set is totally separated.

Therefore $A$ is not a connected set.

Hence the result from definition of a locally connected space.