Cantor Space is not Locally Connected

Theorem

Let $T = \struct {\CC, \tau_d}$ be the Cantor space.

Then $T$ is not locally connected.

Proof

Let $\BB$ be a basis of $T$.

Let $A \in \BB$.

By definition of $\BB$, $A$ is an open set of $T$.

But the Cantor Space is Totally Separated.

Therefore $A$ is not a connected set.

Hence the result from definition of a locally connected space.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $29$. The Cantor Set: $9$